Question

A pair of dice is rolled seven times. Find the probability of rolling an even total exactly once. The probability that an even total is rolled exactly once is (Round to three decimal places as needed.)

Answer 1

the probability of getting an even number total on any one roll of the pair of dice is 0.5

the probability of not getting an even number total on any one roll of the pair of dice is therefore also equal to 0.5.

p = probability of success = 0.5
q = probability of failure = 1 - p = 0.5

The formula for binomial probability is:
p(x) = p^x * q^(n-x) * c(n,x)

in your problem:

x = 1
n = 7
p = 0.5, q = 0.5
n-x = 7-1 = 6

p(x) = p^x * q^(n-x) * c(n,x) becomes:
p(1) = 0.5^1 * 0.5^3 * c(7,1)
c(7,1) = 7! / (1! * 6!) = (7*6!) / (6!) = 7

this becomes:
p(1) = 0.5^1 * 0.5^6 * 7 which becomes:
p(1) = 0.5 * 0.0156 * 7 which becomes:
p(1) = 0.0547

The probability of rolling an even total is rolled exactly once is 0.0547