GATE CS 2016 Set 1 Q.39 (CS Q.29): Probability and Statistics

Consider the following experiment.
Step 1. Flip a fair coin twice.
Step 2. If the outcomes are (TAILS, HEADS) then output Y and stop.
Step 3. If the outcomes are either (HEADS, HEADS) or (HEADS, TAILS), then output N and stop.
Step 4. If the outcomes are (TAILS, TAILS), then go to Step 1.

The probability that the output of the experiment is Y is (up to two decimal places) _____.

Your answer

Official answer and solution

Accepted answers: 0.33 to 0.34
From: Official GATE 2016 answer key

Solution

Set up the recursive probability equation based on coin toss outcomes in the first round.

Let $P(Y)$ be the probability that the experiment ultimately terminates with output $Y$ .

In Step 1, a fair coin is tossed twice. The sample space consists of 4 equally likely outcomes, each with probability $1/4$ :

$(T, H)$ : Outputs $Y$ and stops immediately (probability $1/4$ ).
$(H, H)$ or $(H, T)$ : Outputs $N$ and stops immediately (probability $2/4 = 1/2$ ).
$(T, T)$ : Returns to Step 1, restarting the experiment (probability $1/4$ ).

By the law of total probability:

P(Y) = \frac{1}{4}(1) + \frac{1}{2}(0) + \frac{1}{4} P(Y)

Solve for $P(Y)$ :

P(Y) - \frac{1}{4} P(Y) = \frac{1}{4} \implies \frac{3}{4} P(Y) = \frac{1}{4} \implies P(Y) = \frac{1}{3} \approx 0.3333

Rounded to two decimal places, the probability is 0.33.

Final answer

0.33

Common trap

Omitting the restart on $(T, T)$ and simply taking $1/4 = 0.25$ .

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2016 Computer Science and Information Technology (CS) Session 1
Question: 39 of 65, printed as CS Q.29
Section: Computer Science and Information Technology

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