JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Probability

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A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to probability of getting 9 heads, then the probability of getting 2 heads is :

(A) $\frac{15}{2 ^{14}}$
(B) $\frac{15}{2 ^{12}}$
(C) $\frac{15}{2 ^{8}}$
(D) $\frac{15}{2 ^{13}}$ ✓ Correct

Answer: (D)

Solution

Let the coin be tossed

n

times. For a fair coin,

P (X = r) = (r n) (\frac{1}{2})^{n}

.

Given

P (X = 7) = P (X = 9)

, so

(7 n) = (9 n)

.

This implies

n = 7 + 9 = 16

(since

(r n) = (n - r n)

).

Now,

P (X = 2) = (2 16) (\frac{1}{2})^{16} = \frac{16 \times 15}{2} \cdot \frac{1}{2 ^{16}} = \frac{120}{2 ^{16}} = \frac{120}{65536} = \frac{15}{8192} = \frac{15}{2 ^{13}}

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Single correct option

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