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UPSC NDA II 2015 Mathematics Paper

Question : 19 of 120

Marks: +1, -0

Solution:

Concept:

The probability of exactly

r

tails in

n

coin tosses is given by the binomial formula:

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

, where

p

is the probability of tail,

q = 1-p

Explanation:

We toss a fair coin 5 times, so

n=5

p=\frac{1}{2}

q=1-\frac{1}{2}=\frac{1}{2}

We want the probability that tails appear an odd number of times, i.e., 1, 3, or 5 tails.

Compute each case:

P(1 \text{ tail}) = \binom{5}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{4} = 5 \cdot \frac{1}{32} = \frac{5}{32}

P(3 \text{ tails}) = \binom{5}{3} \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{2} = 10 \cdot \frac{1}{32} = \frac{10}{32}

P(5 \text{ tails}) = \binom{5}{5} \left(\frac{1}{2}\right)^5 \left(\frac{1}{2}\right)^{0} = 1 \cdot \frac{1}{32} = \frac{1}{32}

Sum these probabilities:

\frac{5}{32} + \frac{10}{32} + \frac{1}{32} = \frac{16}{32} = \frac{1}{2}

Answer:

\frac{1}{2}

, which corresponds to option A.

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