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.

Explanation

Multiple of 2 from 1 to 1000 are 2, 4, 6, 8,…., 1000
Let n be the number of terms of above series.
`therefore" ""nth term"=1000`
`rArr " "2+(n-1)2=1000`
`rArr " "2+2n-2=1000`
`rArr " "2n=1000`
`therefore " "n=500`
Since, the number of multiple of 2 are 500.
So, the multiple of 9 are 9, 18, 27,....,999
Let m be the number of term in above series.
`therefore" ""mth term"=999`
`rArr " "9+(m-1)9=999`
`rArr " "9+9m-9=999`
`rArr " "9m=999`
`therefore " "m=111`
Since, the number of multiple of 9 are 111. So, the multiple of 2 and 9 both are 18, 36,..., 990
Let p be the number of terms in above series.
`therefore" ""pth term"=990`
`rArr " "18+(p-1)18=990`
`rArr " "18+18p-18=990`
`rArr " "18p=990`
`therefore " "p=(990)/(18)=55`
Since, the number of multiple of 2 and 9 are 55.
`therefore" Number of miltiple of 2 or 9 = 500 + 111-55 = 556"`
`therefore"Required probability" = (n(E))/(n(S))=(556)/(1000)=0.556`