Assuming seven standard dice are rolled, what is the probability

xonycutieoxl1

xonycutieoxl1

Answered question

2022-06-15

Assuming seven standard dice are rolled, what is the probability their sum equals 17? Show a general approach to solving this problem analytically, using conditional probability, combinatorics, etc

Answer & Explanation

Zayden Andrade

Zayden Andrade

Beginner2022-06-16Added 22 answers

We have k = 7 dice, each takes values in A = { 1 , 2 , 3 , 4 , 5 , 6 }. We want, for n = 17, to know how many solutions c A ( n , k ) of (**) X 1 + + X k = nthere are, where X i A; or rather, what the probability of this event is. We can find c A ( n , k ) by establishing the identity c A ( n , k ) = m A c A ( n m , k 1 ) .(Each solution of (**) must end with X k taking on one value m A and then X 1 + + X k 1 must sum to n m (note, for the initial conditions, that c A ( n , 1 ) = 1 iff n A). Then, the probability P [ X 1 + + X k = n ] is
c A ( n , k ) 6 k
since there are in total 6 k values that the variables X 1 , , X k can take on. Plugging in numbers, we find that your desired probability is
6538 279936 = 2.33... %

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