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प्रश्न
Suppose an integer from 1 through 1000 is chosen at random, find the probability that the integer is a multiple of 2 or a multiple of 9.
उत्तर
We have multiples of 2, from 1 to 1000 are 2, 4, 6, 8, ..., 1000
Let n be the number of terms
a = 2, d = 2, an = 1000
an = a + (n – 1)d
1000 = 2 + (n – 1)2
⇒ 1000 = 2n
⇒ n = 500
Now multiple of 9 from 1 to 1000 are 9, 18, 27, ..., 999
Here a = 9, d = 9 and am = 999 .....[m is the number of terms]
am = a + (m – 1)d
⇒ 999 = 9 + (m – 1)9
⇒ 999 = 9m
⇒ m = 111
Now number multiples of 2 and 9 are 18, 36, 54, ..., 990
Here a = 18, ap = 990, d = 18 .....[p is the number of terms]
∴ ap = a + (p – 1)d
990 = 18 + (p – 1)18
⇒ 990 = 18 + 18p – 18
⇒ 18p = 990
⇒ p = 55
∴ Number of multiples of 2 or 9 = 500 + 111 – 55
= 556
∴ Required probability = `(P(E))/(n(E))`
= `556/1000`
= 0.556
Hence, the required probability = 0.556
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