Question

Example - Draw a Venn diagram with two events A and B where A has 6 outcomes and B has 9 outcomes such that 2 outcomes lie in both events. The sample space has 20 equally likely outcomes in total.

If one of the 20 equally likely outcomes in the sample space is randomly selected, find each of the following probabilities.

1. A occurs.

2. A does not occur.

3. Both A and B occur.

4. A but not B occurs.

5. B but not A occurs.

6. Neither A nor B occurs.

7. A or B occur. Done first by direct count, then by addition rule.

Answer 1

P(A) = n(E)/n(S)

Where P(A) is the probability of an event A

n(E) is the number of favorable outcomes

n(S) is the total number of events in the sample space.  

1.

A occurs have 6 cases.

P = 6/20 = 0.3

2.

P(A doesn't occur) = 1 - P(A occurs) = 1 - 0.3 = 0.7

3.

P(Both A and B) = 2/20 = 0.1

4.

P(A but not B occurs) or P(only A) = 4/20 = 0.2

Only A has 4 cases.

5.

P(B but not A occurs) or P(only B) = 7/20 = 0.35

Only B has 7 cases.

6.

P(Neither A nor B) = 7/20 = 0.35

7 such cases.

7.

P(A or B) = P(A) + P(B) - P(A and B) = 6/20 + 9/20 - 2/20 = 13/20 = 0.65

A or B means either A or B, there will be 13 such cases(4 for only A, 7 for the only B and add 2 that are common).

P = 13/20 = 0.65