Question

Two fair dice are thrown and the number showing on each is noted.
b) Find P(at least one die showing a 3)

Answer 1

(P[X1=3] and P[X2!=3]) or (P[X1!=3 and P[X2=3]) or (P[X1=3] and P[X2=3])

P[X1=3] = 1/6, P[X1!=3] = 5/6, so do you think you can work it out from here.

Note that P(X1 and X2) = P(X1)P(X2), since X1 and X2 are independent.

Answer 2

A more intuitive way to solve this problem is to draw the tree. I can show you how if desired.

Answer 3

Another way to solve the problem is as follows:

The sample space has 36 possible combinations of numbers. These can be set out in column form as follows:
6,6 5,6 4,6 3,6 2,6 1,6
6,5 5,5 4,5 3,5 2,5 1,5
6,4 5,4 4,4 3,4 2,4 1,4
6,3 5,3 4,3 3,3 2,3 1,3
6,2 5,2 4,2 3,2 2,2 1,2
6,1 5,1 4,1 3,1 2,1 1,1
Count the number of occurrences where only one of the dice shows a 3. Then add the one occurrence where both dice show 3. Divide this total by 36 to find the required probability.