Question

I want to see the thought process behind this so that I could approach similar problems on my own, so please attempt to show every step and formula used, thank you.

11.Two fair six sided dice are rolled (i) What is the probability that the smallest of the two results is 3 given that the sum (ii) What is the probability that the sum of the two results is at most 5 given that the ii) What is the probability that the sum of the two results is 7 given that exactly one of the two results is 8? number 2 appeared at least once? of the two results is odd?

Answer 1

a)

P(smallest is 3 | sum is 8)

P(A |B) = P(A and B)/P(B)

now

A = smallest is 3 and B is sum is 8

event A and B is

(2,6),(3,5),(4,4),(5,3),(6,2)

total number = 5

3 is smallest in 2 cases   {(3,5),(5,3)}

hence

P(smallest is 3 | sum is 8)

= P(smallest is 3 and sum is 8)/ P(Sum is 8)

= 2/5

b)

P(sum <= 5 | 2 has appeared at least once)

= P(sum <= 5 and 2 has appeared at least once)/P(2 has appeared at least once)

there are 11 possibilities when 2 has appeared at least once

sum <= 5 and 2 is there at least once

(1,2),(2,1),(2,2),(2,3),(3,2)

total 5 cases

hence

probability = 5/11

c)

sum is 7 | exactly one is odd

sum is 7 and exactly one is odd =

(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)

total 6 cases

exactly one is odd = 18 cases

hence

6/18

=1/3

Please rate