GATE CS 2021 Set 2 Q.32 (CS Q.22): Probability and Statistics

For a given biased coin, the probability that the outcome of a toss is a head is 0.4. This coin is tossed 1,000 times. Let $X$ denote the random variable whose value is the number of times that head appeared in these 1,000 tosses. The standard deviation of $X$ (rounded to 2 decimal places) is ______.

Your answer

Official answer and solution

Accepted answers: 15 to 16
From: Official GATE 2021 answer key

Solution

Identify the distribution of $X$ as Binomial and use the standard deviation formula $\sigma = \sqrt{n p (1-p)}$ .

Each toss is an independent Bernoulli trial with probability of getting head $p = 0.4$ and probability of getting tail $q = 1 - p = 0.6$ .

The number of heads $X$ in $n = 1000$ tosses follows a Binomial distribution: $X \sim \text{Binomial}(n=1000, p=0.4)$ .

The variance of $X$ is:

\sigma^2 = n p q = 1000 \times 0.4 \times 0.6 = 240

The standard deviation $\sigma$ is:

\sigma = \sqrt{240} \approx 15.4919...

Rounding to two decimal places gives $15.49$ (accepted range: 15 to 16).

Final answer

15.49

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

Source

Paper: GATE 2021 Computer Science and Information Technology - Session 2 (CS2)
Question: 32 of 65, printed as CS Q.22
Section: Computer Science and Information Technology

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