Question

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

• A. $\frac{5}{9}$
• B. $\frac{1}{3}$
• C. $\frac{4}{7}$
• D. $\frac{3}{8}$

Hint:

When calculating the expectation of an event involving multiple independent trials (like rolling dice), the total expectation is simply the sum of the individual expectations. For each die, the expectation is the probability of getting the event (in this case, rolling a four), and since the dice rolls are independent, we can just add the expectations together.

Answer 1

The Correct Option is B

The probability of any single die showing a four is \( \frac{1}{6} \). The expected value of \( X \), representing the total count of fours, is calculated by summing the individual expectations for each die:

\[ E(X) = E(X_1) + E(X_2), \]

given that:

\[ E(X_1) = \frac{1}{6}, \quad E(X_2) = \frac{1}{6}. \]

Therefore:

\[ E(X) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}. \]