In a 20-question true-false examination suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answer ‘true’; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.

Given that, there are 20 questions of true-false exam.

If the coin falls head, he answers “true.”

If the coin falls tail, he answers “false.”

Let p be the probability of a correct answer.

That is, p = getting a head and a right answer to be “true” for a question or getting a tail and a right answer to be “false” for a question.

Then, q is the probability of the answer to be incorrect.

And, p + q = 1

⇒ q = 1 – p

Let X denote a random variable representing the number of correct answers out of 20 questions.

Then, the probability of getting r correct answers out of n answered questions is given by,

P (X = r) = ^{n}C_{r}p^{r}q^{n-r}

Here, n = 20 [∵ there are 20 questions in total]

Putting values of n, p, and q in the above equation, we get

…(i)

We need to find the probability that he answers at least 12 questions correctly.

This can be represented as,

Probability = P (X ≥ 12)

It can be written as,

P (X ≥ 12) = P (X = 12) + P (X = 13) + P (X = 14) + P (X = 15) + P (X = 16) + P (X = 17) + P (X = 18) + P (X = 19) + P (X = 20) …(ii)

Put r = 12, 13, 14, 15, 16, 17, 18, 19, 20 in equation (i) subsequently and substitute in (ii), we get

∴, we have got the required probability.

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