P4: Mutually Exclusive Events

Are these events mutually exclusive?

a) Rolling a 3 and rolling a 5 on a dice

b) Drawing a red card and drawing a King from a deck

Solutions:

a) YES - You cannot roll 3 and 5 on one throw

b) NO - You can draw the King of Hearts or King of Diamonds

A fair dice is rolled. Find P(rolling a 2 or a 5).

Solution:

These are mutually exclusive (can't roll both at once)

P(2) = 1/6 and P(5) = 1/6

P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3

A bag contains 3 red, 4 blue and 5 green balls. Find P(red or green).

Solution:

Total balls = 3 + 4 + 5 = 12

P(red) = 3/12 and P(green) = 5/12

P(red or green) = 3/12 + 5/12 = 8/12 = 2/3

Alternative: 8 favourable balls out of 12 total = 8/12

A spinner has 4 sections: Red, Blue, Yellow, Green. These events are exhaustive. Find P(Green).

P(Red) = 0.2, P(Blue) = 0.35, P(Yellow) = 0.25

Solution:

Since events are exhaustive: P(R) + P(B) + P(Y) + P(G) = 1

0.2 + 0.35 + 0.25 + P(G) = 1

0.8 + P(G) = 1

P(Green) = 0.2

In a class of 30: 18 play football, 12 play tennis, 5 play both. Find P(student plays football or tennis).

Solution:

P(Football) = 18/30 = 0.6

P(Tennis) = 12/30 = 0.4

P(Both) = 5/30 ≈ 0.167

P(Football or Tennis) = 0.6 + 0.4 - 0.167 = 0.833

Check: (18 + 12 - 5)/30 = 25/30 ≈ 0.833 ✓

For a dice: Let A = {even numbers} = {2, 4, 6} and B = {factors of 6} = {1, 2, 3, 6}

Find A ∩ B and A ∪ B

Solution:

A ∩ B = {2, 6} (numbers in both sets)

A ∪ B = {1, 2, 3, 4, 6} (all numbers in either set)