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 are: 100000/2=50000 in number

Number of even integers which contain 2 at least once can be counted as:

Case 1. Last digit is 2. SO number is even and contains 2.

So other 4 digits can be varied as 10 possible digits for each place so:10^4 ways

Case 2. Last digit being:0,4,6,8 so 4 ways

So choose one digit out of other 4 in 4 ways and put 2 in it and remaining 3 digits can be varied in:10^3 ways

So,

10^4+4*4*10^3=26000

So required probability is: 26000/50000=0.52

(b)

Number of integers without 2 are:

9^5=59049

Number of integers with 2 atleast once are:100000-59049=40951

Hence required probability is: 26000/40951=0.635