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GO Classes DPP | GATE DA | Probability | Discrete Random Variable

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1 1 vote

A biased die is rolled repeatedly. On each roll, the probability of getting a $6$ is $p$, where $0 < p < 1$. Let $X$ be the random variable denoting the number of rolls until the first $6$ appears, including the roll on which the first $6$ appears. Assume the rolls are independent. Then the expected value of $X$ is

  1. $\dfrac{1}{p}$
     
  2. $\dfrac{1}{1-p}$
     
  3. $\dfrac{1}{p^2}$
     
  4. $\dfrac{1}{1-p^2}$

2 Answers

1 1 vote

To get the first $6$ on the $n$th roll, the first $n-1$ rolls must not be $6$, and the $n$th roll must be $6$.

So, $P(X=n) = (1-p)^{n-1}p$, for $n=1,2,3,\dots$

$\therefore E[X] = \sum_{n=1}^{\infty} nP(X=n)$

$= \sum_{n=1}^{\infty} n(1-p)^{n-1}p$

This is the expectation of a geometric random variable with parameter $p$. We know that

$E[X] = \frac{1}{p}$

Thus, the expected number of rolls required to get the first $6$ is $\boxed{\frac{1}{p}}$

So the correct option is A.

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