Two dice are thrown simultaneously. If X denotes the number of fours, then the expectation of X will be:

The correct option is 1/3.

Let us find the expectation of X, where X denotes the number of fours obtained when two fair dice are thrown simultaneously.

When a single fair die is rolled, the probability of getting a four is:
p = 1 6

The probability of not getting a four is:
q = 1 - p = 1 - 1 6 = 5 6

Since two dice are thrown, the number of trials is n = 2. The random variable X can take the values 0, 1, or 2.
We can find the probability distribution of X as follows:

1. The probability that no die shows a four (X = 0):
P ( X = 0 ) = q × q = 5 6 × 5 6 = 25 36

2. The probability that exactly one die shows a four (X = 1):
P ( X = 1 ) = 2 × p × q = 2 × 1 6 × 5 6 = 10 36

3. The probability that both dice show a four (X = 2):
P ( X = 2 ) = p × p = 1 6 × 1 6 = 1 36

The mathematical expectation of X, denoted as E(X), is calculated using the formula:
E ( X ) = ∑ x i · P ( X = x i )

Substituting the values into the formula:
E ( X ) = 0 × 25 36 + 1 × 10 36 + 2 × 1 36

Simplifying the expression:
E ( X ) = 0 + 10 36 + 2 36 = 12 36 = 1 3

Alternatively, since each roll is an independent Bernoulli trial with a success probability of p = 1/6, the random variable X follows a Binomial distribution with parameters n = 2 and p = 1/6. The expectation of a Binomial distribution is given by:
E ( X ) = n · p = 2 × 1 6 = 1 3