JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Probability

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A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

(A) $\frac{5}{7}$ ✓ Correct
(B) $\frac{2}{7}$
(C) $\frac{3}{7}$
(D) $\frac{5}{6}$

Answer: (A)

Solution

Assign equal prior likelihood to the bag holding

2, 3, 4, 5

6

black balls, since at least two are needed for the observed draw. Given

k

black balls, the chance that both drawn are black is

\frac{( 2 k )}{( 2 6 )}

. By Bayes' theorem the required probability is

\frac{( 2 5 ) + ( 2 6 )}{( 2 2 ) + ( 2 3 ) + ( 2 4 ) + ( 2 5 ) + ( 2 6 )} = \frac{10 + 15}{1 + 3 + 6 + 10 + 15} = \frac{25}{35} = \frac{5}{7}

JEE Main · 2023 · 31 Jan 2023 · Shift 1 · Single correct option

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