JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Probability XII

Let |X| denote the number of elements in set X. Let

S = {1, 2, 3, 4, 5, 6}

be a sample space, where each element is equally likely to occur. If A and B are indepenent events associated with S, then the number of ordered pairs (A,B) such that

1 \leq ∣ B ∣ < ∣ A ∣

, equals

Answer: 422

Solution

For independence,

P (B ∣ A) = P (B)

, which gives

\frac{n ( A \cap B )}{n ( A )} = \frac{n ( B )}{6}

. This requires

n (A)

to have 2 or 3 as prime factors, so

n (A) \in {2, 3, 4, 6}

. With

n (B) < n (A)

n (A) = 2

is excluded. For

n (A) = 3

n (B) = 2

n (A \cap B) = 1

, giving

^{6} C_{4} \times \frac{4 !}{2 !} = 180

pairs. For

n (A) = 4

n (B) = 3

n (A \cap B) = 2

, giving

^{6} C_{5} \times \frac{5 !}{2 ! 2 !} = 180

pairs. For

n (A) = 6

n (B) \in {1, 2, 3, 4, 5}

, giving

2^{6} - 2 = 62

pairs. Total:

180 + 180 + 62 = 422

JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Numerical answer

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JEE Advanced · 2019 · 27 May 2019 · Paper 2 · Probability XII

Solution

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