A fair die is rolled 8 times. What is the probability that an even number (2, 4, or 6) will occur between 2 and 4 times?
A) .2188
B) .4922
C) .6016
D) .8204

Question

Quizplus · Accepted Answer

To solve this problem, we can use the binomial distribution. Let X be the number of times an even number appears in 8 rolls of a fair die. Then, X follows a binomial distribution with parameters n = 8 and p = 1/2 (since the probability of rolling an even number is 3/6 = 1/2). We want to find P(2 <= X <= 4). We can calculate this using the cumulative distribution function (CDF) of the binomial distribution:
P(2 <= X <= 4) = P(X <= 4) - P(X <= 1) = F(4) - F(1)
where F(k) is the CDF evaluated at k.
Using a calculator or software, we find that F(1) ≈ 0.1094 and F(4) ≈ 0.7110. Therefore, P(2 <= X <= 4) ≈ 0.7110 - 0.1094 = 0.6016. Thus, the answer is C.

A Fair Die Is Rolled 8 Times