1- What is the probability of an odd number of sixes turning up in a random toss of 10 fair dice?

2-What is the probability of an even number of sixes turning up in a random toss of 10 fair dice?

3-Are the two numbers equal? Why, or why not?

Thank you for helping.

Guest Aug 18, 2018

#1**+1 **

1- What is the probability of an odd number of sixes turning up in a random toss of 10 fair dice?

We can either have 1 - 3 - 5 - 7 or 9 6s

We have a (1/6) probability of a six on any roll and a (5/6) probability of "not 6" on any roll...so....

P( one 6) = C(10,1)(5/6)^9 (1/6) = 9765625/30233088 ≈ 32.3%

P(three 6s) = C(10,3) (5/6)^7 ( 1/6)^3 = 390625/2519424 ≈ 15.5%

P ( five 6s) = C(10,5)(5/6)^5 (1/6)^5 = 21875/1679616 ≈ 1.3%

P ( seven 6s) = C(10,7) (5/6)^3 (1/6)^7 = 625/2519424 ≈ .02%

P ( nine 6s) = C(10,9)(5/6) (1/6)^9 = 25/30233088 ≈ 0%

Adding all the fractions, we get 58025/118098 ≈ 49.1%

2-What is the probability of an even number of sixes turning up in a random toss of 10 fair dice?

We can have 2 - 4 - 6 - 8 or 10 6s

P(two 6s) = C(10,2) (5/6)^8 (1/6)^2 = 1953125/6718464 ≈ 29.1%

P(four 6s) = C(10,4) (5/6)^6(1/6)^4 = 546875/10077696 ≈ 5.42%

P( six 6s) = C(10,6) (5/6)^4 (1/6)^6 = 21875/10077696 ≈ .21%

P(eight 6s) = C(10,8) (5/6)^2(1/6)^8 = 125/6718464 ≈ 0%

P (ten 6s) = (1/6)^10 = 1/60466176 ≈ 0%

Adding all these fractions, we get 20991751/60466176 ≈ 34.7%

They are not equal....notice that P( one 6 ) > P (two 6s) and P ( three 6s) > P (four 6s) and P( five 6s) > P (six 6s)...etc.

Thus it appears that P(2n - 1 6s) > P(2n 6s) for n = 1 - 5

CPhill
Aug 18, 2018

#2**0 **

This problem appears in a Mathematical Site on the Internet. It is classified under "Probability problem #113." Both the problem and the solution are posted there. Here is the main site:

**http://www.cut-the-knot.org/content.shtml**

Note: The small error in CPhill's counting of EVEN sixes neglected to take "0" into account as follows: C(10,0)*(5/6)^10*(1/6)^0 =0.1615056......., which would give a total of: 0.50867....etc., so that both odd sixes and even sixes must ADD up to 1.

Guest Aug 18, 2018

edited by
Guest
Aug 18, 2018