JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Probability

A number is chosen at random from the set

{1, 2, 3, \dots, 2000}

. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7 . Then the value of 500 p is ____.

Answer: 214

Solution

Let A = set of multiples of 3 in {1, 2, ..., 2000}. Then

n (A) = ⌊ 2000/3 ⌋ = 666

.

Let B = set of multiples of 7 in {1, 2, ..., 2000}. Then

n (B) = ⌊ 2000/7 ⌋ = 285

A \cap B

= set of multiples of 21 in {1, 2, ..., 2000}. Then

n (A \cap B) = ⌊ 2000/21 ⌋ = 95

.

By inclusion-exclusion:

n (A \cup B) = n (A) + n (B) - n (A \cap B) = 666 + 285 - 95 = 856

Probability

p = \frac{856}{2000}

500 p = 500 \times \frac{856}{2000} = \frac{856}{4} = 214

JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Integer answer

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