Operations on events and equally likely outcomes
Answers are for parents and teachers.
1
S = {1, 2, …, 16}, A = “a square number” and B = “even”. Find A ∩ B, A − B, the number of outcomes in A ∪ B and the number in (A ∪ B)′.
A = {1, 4, 9, 16}, so A ∩ B = {4, 16} and A − B = {1, 9}. B has 8 outcomes, so n(A ∪ B) = 4 + 8 − 2 = 10 and (A ∪ B)′ = {3, 5, 7, 11, 13, 15} has 6 outcomes.
2
Let P = “a pupil plays football” and V = “a pupil plays volleyball”. Write in symbols: (a) plays at least one of the two; (b) football only; (c) exactly one of the two; (d) plays at most one.
(a) P ∪ V; (b) P − V = P ∩ V′; (c) (P − V) ∪ (V − P); (d) (P ∩ V)′ = P′ ∪ V′.
3
Two dice are thrown. A = “the sum is 3”, B = “the product is 6” and C = “the difference is 5”. Which pairs of these events are mutually exclusive?
A = {(1, 2), (2, 1)}, B = {(1, 6), (6, 1), (2, 3), (3, 2)}, C = {(1, 6), (6, 1)}. A and B have nothing in common, nor do A and C: both pairs are exclusive. B and C share (1, 6) and (6, 1), so they are not.
4
A box holds 24 cards: 10 show a circle, 8 a square and 6 a triangle. One card is drawn at random. Find P(circle), P(not a square) and P(circle or triangle).
P(circle) = 10/24 = 5/12; P(not a square) = 16/24 = 2/3; P(circle or triangle) = 16/24 = 2/3.
5
Two dice are thrown. Find the probability that the first die shows a larger number than the second.
Of the 36 pairs, 6 have equal numbers. The other 30 split equally between “first larger” and “second larger”, so 15 pairs and P = 15/36 = 5/12.