A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500 is ________. [2021]

Question

A number is chosen at random from the set {1, 2, 3, ..., 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 ________. [2021]

Smart Achievers · Accepted Answer

(214)

$E = set of numbers divisible by 3$

$E = {3, 6, 9, 12, \dots, 1998}$

$∴$ $n (E) = 666$

$F = set of numbers divisible by 7$

$F = {7, 14, 21, \dots, 1995}$

$∴$ $n (F) = 285$

$E \cap F = set of numbers divisible by 3 and 7$ $= {21, 42, \dots, 1995}$

$∴$ $n (E \cup F) = 666 + 285 - 95 = 856$

$Required probability = \frac{856}{2000} = P$

$So, 500 P = \frac{856}{2000} \times 500 = 214$

Solution

Q.
A number is chosen at random from the set {1, 2, 3, ..., 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 ________. [2021]

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Solution

Q. A number is chosen at random from the set {1, 2, 3, ..., 2000}. Let p be the probability that the chosen number is a multiple of 3 or a multiple of 7. Then the value of 500p is ________. [2021]

Ans. (214) E= set of numbers divisible by 3 E={3,6,9,12,…,1998} ∴ n(E)=666 F= set of numbers divisible by 7 F={7,14,21,…,1995} ∴ n(F)=285 E∩F= set of numbers divisible by 3 and 7={21,42,…,1995} ∴ n(E∪F)=666+285-95=856 Required probability=8562000=P So, 500P=8562000×500=214

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Q.
A number is chosen at random from the set {1, 2, 3, ..., 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 ________. [2021]