Question

the numbers 1,2,3,4, and 5 are written on slips of paper, and 2 slips are drawn at random one at a time without replacet. find the probability that the first number is 3, given that the sum is 4

Answer 1

Consider such events:

A - slip with number 3 is chosen;

B - the sum of numbers is 4.

You have to count
`Pr(A|B).`

Use formula for conditional probability:

`Pr(A|B)=(Pr(A\cap B))/(Pr(B)).`

1. The event
`A\cap B` consists in selecting two slips, first is 3 and second should be 1, because the sum is 4. The number of favorable outcomes is exactly 1 and the number of all possible outcomes is 5·4=20 (you have 5 ways to select 1st slip and 4 ways to select 2nd slip). Then the probability of event
`A\cap B` is

`Pr(A\cap B)=(1)/(20).`

2. The event
`B` consists in selecting two slips with the sum 4. The number of favorable outcomes is exactly 2 (1st slip 3 and 2nd slip 1 or 1st slip 1 and 2nd slip 3) and the number of all possible outcomes is 5·4=20 (you have 5 ways to select 1st slip and 4 ways to select 2nd slip). Then the probability of event
`B` is

`Pr(B)=(2)/(20)=(1)/(10).`

3. Then

`Pr(A|B)=((1)/(20) )/((1)/(10) )=(1)/(2).`

Answer:
`(1)/(2).`