Question

A pair of dice is thrown two times. If $X$ represents the number of doublets obtained, then the expectation of $X$ is

• A. $\dfrac{1}{6}$
• B. $1$
• C. $\dfrac{1}{3}$
• D. $\dfrac{11}{36}$

Hint:

When repeated trials are involved with constant probability, use the Binomial distribution to find expected value: $E[X] = np$.

Answer 1

The Correct Option is C

A doublet occurs when both dice display the same number, e.g., (1,1), (2,2), ..., (6,6).
There are 6 such outcomes from a total of $6 \times 6 = 36$ possible outcomes when rolling a pair of dice once.
Consequently, the probability of obtaining a doublet in a single trial is $\dfrac{6}{36} = \dfrac{1}{6}$.
Consider the scenario where the dice are thrown twice.
Let $X$ represent the count of doublets acquired in 2 independent trials.
$X$ adheres to a Binomial distribution with parameters $n = 2$ and $p = \dfrac{1}{6}$.
The expected value $E[X]$ is calculated as $n.p = 2.\dfrac{1}{6} = \dfrac{2}{6} = \dfrac{1}{3}$.
Therefore, the anticipated number of doublets is $\dfrac{1}{3}$.