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 is obtained when both dice show the same number, such as (1,1), (2,2), ..., (6,6).
There are 6 such outcomes out of a total of $6 \times 6 = 36$ outcomes when a pair of dice is thrown once.
So, the probability of getting a doublet in one trial is $\dfrac{6}{36} = \dfrac{1}{6}$.
Now, the dice are thrown twice.
Let $X$ be the number of doublets obtained in 2 independent trials.
Then $X$ follows a Binomial distribution with $n = 2$, $p = \dfrac{1}{6}$.
The expectation $E[X] = n.p = 2.\dfrac{1}{6} = \dfrac{2}{6} = \dfrac{1}{3}$.
So, the expected number of doublets is $\dfrac{1}{3}$.