เอกสาร ความน่าจะเป็น v2

1 ความนาจะเปน เบือ้ งตน 1. กฏเกณฑก ารนบั เบอื้ งตน คําชแ้ี จง ใหน ักเรียนพจิ ารณาตัวอยางตอไปน้ี ตัวอยางท่ี 1 ใน Audition Game ผเู ลน จะสามารถแตงตัวใหก ับตัวละครของตนเองไดตามใจชอบ ถา หาก ตวั ละครของเดก็ ชายคนหนง่ึ สามารถเลือกเส้อื ได 3 ตัว และสามารถเลือกกางเกงได 2 ตวั อยากทราบวา ตัวละครของเดก็ คนนี้จะสามารถแตงตัวทงั้ หมดไดกแี่ บบ วธิ ที ํา จากแผนภาพจะเห็นวา ตวั ละครของเด็กชายคนน้ีจะสามารถแตงตวั โดยมเี ส้ือ 3 ตัวและกางเกง 2 ตัว ไดท้ังหมด ......................................แบบ ซงึ่ เทา กบั จํานวนเส้อื ................... (ตัว) คูณ ดวยจํานวนกางเกง...................(ตัว) หรือเทากับ...............×...............= ............... นั่นเอ

1 ความนาจะเปน เบ้อื งตน 1. กฏเกณฑการนบั เบอ้ื งตน คําช้แี จง ใหนกั เรียนพจิ ารณาตวั อยา งตอไปน้ี ตัวอยา งที่ 1 ใน Audition Game ผูเลนจะสามารถแตง ตัวใหกับตัวละครของตนเองไดตามใจชอบ ถา หาก ตวั ละครของเดก็ ชายคนหนึ่ง สามารถเลอื กเส้ือได 3 ตัว และสามารถเลือกกางเกงได 2 ตวั อยากทราบวา ตัวละครของเด็กคนนจ้ี ะสามารถแตง ตวั ทง้ั หมดไดก่แี บบ วธิ ที าํ จากแผนภาพจะเหน็ วา ตวั ละครของเด็กชายคนน้ีจะสามารถแตงตวั โดยมเี ส้ือ 3 ตัวและกางเกง 2 ตวั ไดท งั้ หมด ......................................แบบ ซงึ่ เทา กบั จํานวนเสือ้ ................... (ตัว) คณู ดวยจํานวนกางเกง...................(ตัว) หรอื เทากับ...............×...............= ............... นั่นเอง

2 ตัวอยางท่ี 2 โดเรมอ นมีประตกู าลเวลาทั้งหมด 3 ประตู ถาโดเรมอ นสามารถเดินเขาและเดนิ ออกประตู กาลเวลาใดกไ็ ด จงหาวา โดเรมอนจะมีวธิ ีเดินเขาและออกประตกู าลเวลานไ้ี ดท้งั หมดกว่ี ธิ ี วธิ ที ํา จากแผนภาพจะเห็นวา โดเรมอนจะสามารถเดินเขาและออกประตูวิเศษโดยมี 3 ประตู ไดท ั้งหมด ...................วิธี ซง่ึ เทา กับจํานวนประตูทเ่ี ดนิ เขา ................... (ประตู) คูณ ดว ยจาํ นวนประตูท่ีเดินออก...................(ประตู) หรือเทากับ...............×...............= ............... นั่นเอง จากตัวอยางที่ 1 และ 2 เราสามารถสรปุ เปน กฎ (ในกรณีท่ัวไป) ไดด งั นี้ หลกั การคูณ กฎข้อที่ 1 ..................................................................................................................................................................................................................... ............ ..................................................................................................................................................................................................................................................................

3 ตัวอยางที่ 3 มถี นนจากตาํ บล ก ไปตําบล ข อยู 5 สาย และมีถนนจากตําบล ก ไปตําบล ข อยู 3 สาย อยากทราบวา ถา ตองการเดินทางจากตําบล ก ไปยังตําบล ค โดยผา นตาํ บล ข จะทําไดท้ังหมดกว่ี ิธี วิธที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ตัวอยางท่ี 4 ในการทอดลูกเตา สองลูก จะปรากฎผลไดท้ังหมดกวี่ ิธี วธิ ีทาํ ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ตวั อยา งที่ 5 ทา เรอื ขามแมน ้ําเจาพระยา มีเรือขา มฟาก 5 ลํา ถานักเรยี นคนหน่ึงตองการขา มฟากไปและกลบั จงหาจาํ นวนวิธที น่ี กั เรยี นคนน้จี ะเดนิ ทางขา มฟากโดย 1) ไปและกลับจะลงเรือลาํ ใดกไ็ ด 2) ไปและกลับจะลงเรือซ้าํ ลาํ เดิมไมได 3) ไปและกลบั ลงเรอื ลาํ เดยี วกัน วธิ ีทาํ 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4 โดยการขยายกฎขอท่ี 1 ใหใชก ับการทํางานหรอื ปฏิบัติการมากกวา 2 อยาง จะไดกฎทว่ั ไป ดังน้ี กฎข้อท่ี 2 ....................................................................................................................................................................................................................................... .................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................. ตัวอยางที่ 6 เมื่อโยนเหรียญหนึง่ เหรียญหน่งึ ครัง้ เหรียญอาจข้นึ หวั หรอื กอ ยกไ็ ด อยากทราบวา ถาโยน เหรยี ญหนงึ่ เหรียญ 3 ครง้ั จะไดผ ลลพั ธต างๆกันท้ังหมดก่ีวิธี วธิ ที าํ สมมตใิ หH แทนการโยนเหรียญแลว ข้ึนหัว T แทนการโยนเหรยี ญแลวข้นึ กอย หาผลลพั ธท ้ังหมดโดยใชแผนภาพตน ไมดงั น้ี จากแผนภาพจะเหน็ วา งานนี้แบงออกเปน 3 ข้นั ตอน คอื ขน้ั ตอนท่ี 1 โยนเหรยี ญครง้ั ท่ีหนง่ึ จะได......................................วิธี ขั้นตอนท่ี 2 โยนเหรียญครง้ั ท่ีสองจะได. .....................................วธิ ี ข้นั ตอนท่ี 3 โยนเหรยี ญครัง้ ที่สามจะได......................................วธิ ี ดงั น้นั การโยนเหรียญหนึ่งเหรยี ญ 3 ครงั้ จะไดผ ลลัพธต า งๆกันทัง้ หมด......................................วิธี

5 ตัวอยางที่ 7 รานคาจําหนายเส้ือผาสําเร็จรูป จําหนายเส้ือทั้งหมด 5 แบบ แตละแบบมี 4 สี และแตละสีมี 3 ขนาด ถาตองการจัดเส้อื โชวห นา รานใหครบทกุ สี ทกุ ขนาด จะตอ งเตรยี มไมแขวนเสอ้ื ทงั้ หมดกี่อัน วิธที าํ ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ตัวอยา งที่ 8 ถาตอ งการสรางจาํ นวนเต็มบวกท่มี ีส่ีหลกั จากเลขโดด 0-9 จะสรางไดท ง้ั หมดก่ีจาํ นวน เมื่อ ตอ งการให 1) แตละหลกั ใชเลขซ้ํากนั ได 2) แตล ะหลักใชเ ลขซํ้ากันไมไ ด วธิ ีทํา ......................................................................................................................................................................................................................................................................... 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6 ตัวอยางท่ี 9 จะมวี ธิ ที แ่ี ตกตา งกนั ทงั้ หมดก่ีวิธีทีจ่ ะเลือกสี 8 สี มาทาลกู เตา หนาละสี วธิ ีทาํ ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ตวั อยางท่ี 10 จงหาจาํ นวนวธิ ที ่เี ปน ไปไดท้งั หมดในการสง จดหมาย 4 ฉบบั ลงในตูไปรษณยี  5 ตู ถา ... - จดหมายแตละฉบับจะใสต ใู ดก็ได - หา มใสจดหมายมากกวา 1 ฉบับ ในตเู ดยี วกนั วิธีทํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... .........................................................................................................................................................................................................................................................................

7 แบบฝก ทักษะท่ี 1.1 เรื่อง กฎเกณฑเ บื้องตนเก่ยี วกบั การนบั 1. นักเรยี นคนหน่งึ ตองการเลือกเรยี นภาษาตางประเทศ 1 ภาษา จาก 5 ภาษา และตองการเรียนวิชา วทิ ยาศาสตร 1 วิชาจาก 4 วชิ า นักเรียนคนน้ีจะมวี ธิ เี ลอื กเรียนไดทั้งหมดกีว่ ธิ ี วิธที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 2. นกั เรียนหองหนึ่งมี 40 คน ตอ งการเลือกหัวหนาหองและรองหวั หนาหอ ง จะมีวธิ เี ลือกไดท งั้ หมดกี่วธิ ี วธิ ที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 3. มีถนนจากเมอื ง ก ไปยงั เมอื ง ข 6 สาย และจากเมอื ง ข ไปยงั เมือง ค 4 สาย จงหาจาํ นวนวิธีท้งั หมด ท่จี ะขับรถยนต - จากเมือง ก ไปยังเมือง ค - จากเมือง ก ไปยังเมือง ค แลวขับกลบั - จากเมืองก ไปยังเมือง ค แลว ขบั กลับโดยไมขบั ผา นซา้ํ ในถนนสายเดยี วกนั วธิ ที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... .........................................................................................................................................................................................................................................................................

8 ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 4. ถา นําเลขโดด 0-4 มาสรา งเปนจํานวนที่มีสองหลกั จะสรา งไดท้งั หมดก่ีจาํ นวน ถา - ไมม ีเงื่อนไขใดๆ - แตละหลกั เลขตองไมซํ้ากนั วธิ ที าํ ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 5. มนี กั เรยี น 5 คน ตองการนํานักเรยี นจาํ นวน 3 คนมายนื เรียงเปนแถว จงหาจํานวนวธิ ีทจี่ ะจดั ได ท้งั หมด วธิ ีทํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... .........................................................................................................................................................................................................................................................................

9 6. จงนาํ ตวั อักษรจากคําวา SPECIAL มาเรียงเปน คาํ ใหมโดยไมค ํานงึ ถึงความหมาย จะไดค ําไมที่ไมซํ้ากบั คาํ เดิมท้ังหมดก่ีคาํ วธิ ที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 7. ชมรมตนกลาคณิตศาสตร ตองการเลอื กประธาน รองประธาน และเลขานกุ าร ตําแหนง ละ 1 คน จาก ผสู มัครทง้ั หมด 8 คน จะมีวิธเี ลือกประธาน รองประธาน และเลขานุการไดกวี่ ิธี ถา มีผสู มัครหน่ึงคนไม สมคั รใจเปนประธาน วิธที าํ 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......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 8. ขอสอบวชิ าคณติ ศาสตรมคี ําถามแบบถูกผิด อยทู ้ังหมด 8 ขอ ตา งๆ กัน ถาเด็กหญิงเดน จันทร ทํา ขอ สอบวิชานีโ้ ดยการเดา เดก็ หญิงเดนจันทรจ ะมวี ิธีการทําขอ สอบทงั้ 8 ขอ ไดทัง้ หมดกี่วธิ ี วิธีทาํ ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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10 9. นกพริ าบ 5 ตัว จะเลือกบินมาเกาะตนไม 3 ตน ไดก่วี ิธี วธิ ที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 10. การโยนลกู เตาสองลูกทีแ่ ตกตางกนั ใหไดแตมคูทงั้ สองลูกทําไดก ่วี ธิ ี วิธที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 11. ในการทอดลูกเตา n ลูกพรอมกัน 1 ครงั้ จะปรากฏผลไดกี่วิธี วธิ ที ํา 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......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 12. รถยนตคันหนึง่ มีท่ีนัง่ อยขู างหนา (รวมทค่ี นขบั ) 2 ท่ี และขางหลงั 3 ที่ ถามีคนงานทัง้ หมด 7 คน ซึง่ 3 คนขบั รถได จะจัดคนเขาน่ังรถไดก่ีวธิ ี วิธที ํา ......................................................................................................................................................................................................................................................................... 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11 กฎข้อท่ี 3 กฎโดยการบวก ........................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................... ........................... ....................................................................................................................................................................................................................................................... .......................... ตวั อยา งที่ 1 ถาหมายเลขทะเบยี นรถยนตในกรุงเทพมหานครประกอบดว ยพยญั ชนะ 2 ตวั และตวั เลข 1 ถงึ 4 หลกั ตวั อยางเชน กข 1 , กค 12 , กง 123 และ กจ 1234 อยากทราบวา จะมีหมายเลขทะเบียนรถยนตท ี่ แตกตางกนั ท้งั หมดไดกห่ี มายเลข วธิ ที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ตัวอยางท่ี 2 จะจัดชาย 3 คน และหญงิ 3 คนเขาแถวตรงไดก ่ีวิธี โดยใหชายและหญิงยืนสลับกนั วิธที ํา 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12 ตวั อยา งที่ 3 บา นพักและสถานที่ทํางานของเพลงิ ต้ังอยรู มิ แมน ้ําเจาพระยา เพลงิ โดยสารเรือยนตไปทํางาน ตอนเชาและกลบั ที่พกั ในตอนเยน็ ถา เรือยนตม ี 3 มขี นาด คือ ขนาดใหญ 3 ลํา ขนาดกลาง 5 ลาํ และ ขนาด เล็ก 2 ลํา จงหาจาํ นวนวธิ ีทแี่ ตกตางกนั ทัง้ หมดท่ีเพลงิ โดยสานเรอื ยนตไปทํางานและกลบั บานพักดว ยเรือขนาด เดยี วกนั แตไมใชเ รอื ลําเดยี วกัน วิธีทํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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......................................................................................................................................................................................................................................................................... ตัวอยางที่ 4 กาํ หนดใหมีบัตร 4 ใบ เขยี นหมายเลข 1 , 2 ,3 และ 4 นําบตั รเหลาน้ีมาเรียงเปนจํานวนตา งๆ ดังนี้ไดก ีจ่ ํานวน วธิ ีทาํ ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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13 แบบฝกทกั ษะท่ี 1.2 เรอ่ื ง กฎเกณฑเบ้อื งตนเกี่ยวกับการนับ 1. นกั เรยี นกลุมหนงึ่ มเี ดก็ ผชู าย 3 คน และเด็กผหู ญงิ 4 คน จงหาจํานวนวธิ เี ลือกตัวแทน 2 คนท่ีมีเพศ เดยี วกัน วิธที ํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 2. ตองการนําเลขโดด 0-9 มาสรางเปน จาํ นวนเตม็ คบู วก 3 หลกั โดยไมมีเลขในหลกั ใดซา้ํ กันเลย จะ สรางจาํ นวนเต็มบวกไดกี่จํานวน วิธที ํา 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......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 3. กาํ หนดใหใชตัวเลข 0,1,2,3,4,5,6 และ 7 สรางจํานวนท่ีมี 3 หลกั โดยกาํ หนดวา แตล ะหลักไมมี ตัวเลขซํา้ กนั จงหาวาจาํ นวนเลขสามหลักทเ่ี ปน เลขคู มที ้ังหมดกจี่ ํานวน วธิ ีทาํ 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14 4. จากตัวเลข 0, 1 , 2 , 3 , 4 และ 5 จะสรา งจํานวนท่ีมี 3 หลัก และจํานวนเหลา นห้ี ารดว ย 5 ลงตวั จะ มที ั้งหมดกี่จํานวน เม่ือ 4.1 ใชเ ลขซา้ํ ได 4.2 ใชเ ลขซาํ้ ไมได วิธีทํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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......................................................................................................................................................................................................................................................................... 5. จงหาจํานวนคีท่ ่ีอยูระหวา ง 3,000 ถงึ 5,000 มกี ี่จาํ นวนโดยท่ีแตละหลักมีตวั เลขแตกตา งกนั วิธีทํา ......................................................................................................................................................................................................................................................................... ......................................................................................................................................................................................................................................................................... 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