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.

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CBSE | Class 11 | Maths | NCERT Exemplar | Probability

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.

Solution:
We have integers $1,2,3, \ldots 1000$
We have integers $1,2,3, \ldots 1000$
$(S)=1000$
No. of integers that are multiple of $2=500$
Let’s say the no. of integers that are multiple of 9 be $n$ .
The nth term $=999=9+(n-1) 9=999$
$\Rightarrow \quad 9+9 n-9=999$ $\Rightarrow \quad n=111$
From 1 to 1000 , the no. of multiples of 9 is 111 .
The multiple of 2 and 9 both are $18,36, \ldots, 990$ .
Let’s say that $\mathrm{m}$ be the no. of terms in above series.
$\begin{array}{ll} \therefore & & \text { mth term }=990 \\ \Rightarrow & & 18+(\mathrm{m}-1) 18=990 \\ \Rightarrow & & 18+18 \mathrm{~m}-18=990 \\ \Rightarrow & \mathrm{m} & =55 \end{array}$
No. of multiples of 2 or $9=500+111-55=556=n(E)$
$\therefore \quad$ The required probability $=\frac{n(E)}{n(S)}=\frac{556}{1000}=0.556$