A positive integer less than 100000 is picked at random a Gi

A positive integer less than 100,000 is picked at random.

(a) Given that the integer chosen is even, what is the probability that it contains the digit 2 at least once?

(b) Given that the integer chosen contains the digit 2 at least once, what is the probability that it is even?

(a)

Number of even integers is: 100,000/2=50,000

We first count even integers without 2.

Last digit can be picked in 4 ways:0,4,6,8

Now 4 other digits each can take values from:0,1,3,4,... ,9

So 10 for each so ;9^4 combinations=9561 combinations

Hence, 4*9561=26244 integers and including 100000 it becomes: 26245 integers.

So probability of an even digit containing 2 at least once is:

(50000-26245)/50000=0.47512

(b)

Number of even integers which contains 2 at least once is: 50000-26245=23755

Number of integers which contain 2 at least once is: 100000-9^5=40951

Hence required probabilty is: 23755/40951=0.58008351444