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)

P(rain) = 0.3 and P(wind) = 0.6; they can occur together with P(both) = 0.15. Find P(rain or wind).

0.3 + 0.6 − 0.15 = 0.75

M1 for adding, M1 for subtracting the overlap, A1 for 0.75. Forgetting the subtraction double-counts the overlap.

A spinner: P(A) = 0.3, P(B) = 0.45. A and B are mutually exclusive. Find P(neither A nor B).

P(A or B) = 0.75 → P(neither) = 1 − 0.75 = 0.25

M1 for the sum, A1 for the complement. 'Neither' = 1 − P(either), building on the exclusive rule.

Q1. No — the 6 is even, so both can happen together

Q2. 5/10 + 2/10 = 7/10 (mutually exclusive — one colour each)

Q3. 0.4 + 0.35 = 0.75

Q4. 1 − 0.75 = 0.25

Q5. 40 + 35 − 15 = 60 → 0 like neither

Q6. Not exclusive (the queen of hearts). P = 13/52 + 4/52 − 1/52 = 16/52 = 4/13

Q7. 0.5 + 0.4 − 0.2 = 0.7