JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Probability

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Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is

\frac{k}{2 ^{15}}

, then k is equal to :

(A)30
(B)90
(C)15
(D)60✓ Correct

Answer: (D)

Solution

A single roll gives an odd number with probability

\frac{1}{2}

. Equal probabilities for seven and nine odd outcomes require

(7 n) = (9 n)

, so

n = 16

. An even number exactly twice means an odd number fourteen times, with probability

(2 16) (\frac{1}{2})^{16} = \frac{120}{2 ^{16}} = \frac{60}{2 ^{15}}

. Therefore

k = 60

JEE Main · 2023 · 10 Apr 2023 · Shift 2 · Single correct option

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