Question

Let P be the uniform probability on the integers from 1 to 99. Let B be the subset of numbers which have the digit 3. Let A be the subset of even numbers. What is P(A), P(B)? What is P(A|B)? P(B|A)?

Answer 1

We have integers from 1 to 99.

Event A : subset of even numbers

Event B : subset of numbers which have the digit 3

Event A : {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98}

Event B : {3,13,23,30,31,32,33,34,35,36,37,38,39,43,53,63,73,83,93}

Total number in event A = n(A) = 49

Total number in event B = n(B) = 19

so total outcomes are = 99

favourable outcomes for event A is = 49

favourable outcomes for event B is = 19

Now P(A|B) and P(B|A) is a conditional probability and is equal to :

means the intersection value between the two events, that is the common values,

So = {30,32,34,36,38}

there are 5 common values in between them

=>

First finding P(A|B) :

Now finding P(B|A) :

I have rounded the answer to 4 decimal places.