GATE CS 2016 Set 1 Q29 — Probability: Random Variables and Distributions

When we flip a fair coin twice in Step 1, there are four equally likely outcomes, each with a probability of $\frac{1}{4}$:

• $\text{(HEADS, HEADS)}$ with probability $\frac{1}{4}$
• $\text{(HEADS, TAILS)}$ with probability $\frac{1}{4}$
• $\text{(TAILS, HEADS)}$ with probability $\frac{1}{4}$
• $\text{(TAILS, TAILS)}$ with probability $\frac{1}{4}$

Let $P(Y)$ be the probability that the final output of the experiment is $Y$.

Based on the steps of the experiment:

1. If the outcome is $\text{(TAILS, HEADS)}$, we output $Y$ and stop. This occurs with probability $\frac{1}{4}$.
2. If the outcome is $\text{(HEADS, HEADS)}$ or $\text{(HEADS, TAILS)}$, we output $N$ and stop. This occurs with probability $\frac{1}{4} + \frac{1}{4} = \frac{1}{2}$.
3. If the outcome is $\text{(TAILS, TAILS)}$, the experiment restarts from Step 1. This occurs with probability $\frac{1}{4}$.

We can write the recurrence relation for $P(Y)$ as follows:

Solving for $P(Y)$:

Rounding to two decimal places, the probability that the output of the experiment is $Y$ is 0.33.