Question

What is the expected number of even rolls on a fair six-sided die in 30 rolls?

Answer 1

To find the expected number of even rolls when rolling a fair six-sided die 30 times, we start by determining the probability of rolling an even number on a single die roll.

The outcomes for a six-sided die are: 1, 2, 3, 4, 5, and 6. Among these, the even numbers are 2, 4, and 6. Thus, there are 3 even numbers out of 6 possible outcomes.

The probability \( P \) of rolling an even number is:

\[
P(\text{even}) = \frac{\text{number of even outcomes}}{\text{total outcomes}} = \frac{3}{6} = \frac{1}{2}
\]

Next, we define a random variable \( X \) which represents the number of even rolls in 30 rolls. Each roll is an independent event, and the expected number of even rolls can be calculated using the formula for the expected value of a binomial distribution, where \( n \) is the number of trials (rolls) and \( p \) is the probability of success (rolling an even number).

In our case:

- \( n = 30 \) (the number of rolls)
- \( p = \frac{1}{2} \) (the probability of rolling an even number)

The expected value \( E(X) \) is given by:

\[
E(X) = n \times p = 30 \times \frac{1}{2} = 15
\]

Thus, the expected number of even rolls in 30 rolls of a fair six-sided die is:

\[
\boxed{15}
\]