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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 : 115 of 120

Marks: +1, -0

What is the value of n ?

Solution:

Concept:

Use complementary probability to define

p

and

q

in terms of

n

, then solve the equation

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

Explanation:

Step 1: Probability of getting at least one tail in

n

tosses of a fair coin:

p = 1 - P(\text{no tail}) = 1 - \left(\frac{1}{2}\right)^n

Step 2: Probability of getting at least two tails:

q = 1 - [P(\text{no tail}) + P(\text{exactly one tail})]

P(\text{exactly one tail}) = \binom{n}{1}\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)^{n-1} = n \left(\frac{1}{2}\right)^n

Therefore

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

Step 3: Substitute into

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

\left[1 - \frac{1}{2^n}\right] - \left[1 - \frac{1+n}{2^n}\right] = \frac{5}{32}

Simplify:

\frac{1+n}{2^n} - \frac{1}{2^n} = \frac{5}{32}

, so

\frac{n}{2^n} = \frac{5}{32}

Step 4: Solve

\frac{n}{2^n} = \frac{5}{32}

. Since

32 = 2^5

, compare directly:

n = 5

Answer:

n = 5

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