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# Probability problem

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Jack rolls 5 fair six-sided dice. What is the probability that at least three dice show the same number?

Feb 14, 2022

#1
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P ( 3 show the same number )  =   C(5,3) (1/6)^3 ( 5/6) (4/6)   =  25/ 972 = 200/7776

P( 4 show the same number)  = C(5,4) (1/6)^4 ( 5/6)   = 25/7776

P( 5 show the same number ) =  C(5,5) ( 1/6)^5  = 1/7776

Total probability  = (200 + 25 + 1) / 7776  =   226 /7776 = 113/ 3888   Feb 14, 2022
edited by CPhill  Feb 14, 2022
#3
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The probability that exactly 3 dice show the same is $${5 \choose 3} \times {1 \over 6^3} \times{5 \over6^2} \times 6$$. $$5 \choose 3$$ ways to pick the rolls that are the same, $${1 \over 6}^3$$ chance of 3 successes,  $${5 \over 6}^2$$ chance of 2 failures, and 6 different trios of numbers that are the same.

The probability that exactly 4 dice show the same is $${5 \choose 4} \times {1 \over6}^4\times {5 \over6} \times 6$$. With the same logic as above.

The probability that all 5 dice show the same is $${1 \over 6}^5\times6$$

Add everything up, and you get $$\color{brown}\boxed{23\over108}$$

BuilderBoi  Feb 15, 2022
#2
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3 of a kind ==6 *[5 C 3 * 5^2]==1,500
4 of a kind ==6 *[5 C 4 * 5^1]==150
5 of a kind ==6 *[5 C 5 * 5^0]==6

Therefore, the probability of at least 3 showing the same number is: [1,500 + 150 + 6] ==1,656 / 6^5 ==23 / 108

Feb 14, 2022