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UPSC NDA I 2024 Mathematics Paper

Question Numbers: 115-116

Consider the following for the items that follow :

An unbiased coin is tossed n times. The probability of getting at least one tail is p and the probability of at least two tails is q and

p-q=\frac{5}{32}

Question : 116 of 120

Marks: +1, -0

What is the value of p + q ?

Solution:

Concept:

The sum

p+q

is found by simplifying each term individually and then adding the results.

Explanation:

Step 1: Compute $p$ .

Given

p = 1 - \left( \frac{1}{2} \right)^5

Calculate the power:

\left( \frac{1}{2} \right)^5 = \frac{1}{32}

p = 1 - \frac{1}{32} = \frac{31}{32}

Step 2: Compute $q$ .

Given

q = 1 - \left( \frac{1+5}{2^5} \right)

First evaluate the numerator:

1+5 = 6

Then

2^5 = 32

, so

\frac{1+5}{2^5} = \frac{6}{32}

Simplify

\frac{6}{32} = \frac{3}{16}

Thus

q = 1 - \frac{3}{16} = \frac{13}{16}

Step 3: Add $p$ and $q$ .

Convert

\frac{13}{16}

to a denominator of 32:

\frac{13}{16} = \frac{26}{32}

p + q = \frac{31}{32} + \frac{26}{32} = \frac{57}{32}

Answer:

The value of

p+q

\frac{57}{32}

, which corresponds to option A.

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