A fair die is thrown until the number 2 appears. What is the prob… — Solution | JEE Mathematics

MCQMediumJEE 2024Probability Basics

A fair die is thrown until the number $2$ appears. What is the probability that $2$ appears in an even number of throws?

Answer

Correct answer:C

Standard Method

Given: A fair die is thrown until the number $2$ appears.

Find: The probability that $2$ first appears on an even-numbered throw.

The probability of getting $2$ on one throw is $\frac{1}{6}$ , and the probability of not getting $2$ is $\frac{5}{6}$ .

For $2$ to appear on an even throw, it can first appear on the $2^{\text{nd}}$ , $4^{\text{th}}$ , $6^{\text{th}}$ , and so on.

So the required probability is

P(\text{Even}) = \left(\frac{5}{6}\right)\left(\frac{1}{6}\right) + \left(\frac{5}{6}\right)^3\left(\frac{1}{6}\right) + \left(\frac{5}{6}\right)^5\left(\frac{1}{6}\right) + \cdots

Geometric Series Evaluation

This is an infinite geometric series:

P(\text{Even}) = \sum_{k=1}^{\infty} \left(\frac{5}{6}\right)^{2k-1} \times \frac{1}{6}

Using first term and common ratio

The first term is

a = \frac{5}{36}

and the common ratio is

r = \left(\frac{5}{6}\right)^2 = \frac{25}{36}

Final computation

Using the sum of an infinite geometric series,

S = \frac{a}{1-r}

we get

P(\text{Even}) = \frac{\frac{5}{36}}{1-\frac{25}{36}} = \frac{\frac{5}{36}}{\frac{11}{36}} = \frac{5}{11}

Therefore, the correct option is C.

Assuming the probability is only for the $2^{\text{nd}}$ throw. The event includes $2$ appearing first on the $2^{\text{nd}}, 4^{\text{th}}, 6^{\text{th}}$ , and all even-numbered throws. Sum the entire infinite series.
Using common ratio $\frac{5}{6}$ instead of $\left(\frac{5}{6}\right)^2$ . Consecutive valid terms differ by two extra failures before success, so the ratio is $\frac{25}{36}$ .
Forgetting that the first appearance condition requires all previous throws to be not $2$ . For appearance on the $2n^{\text{th}}$ throw, the first $2n-1$ throws must avoid $2$ except the last one, which must be $2$ .

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.