Independent and dependent events
Answers are for parents and teachers.
1
S = {1, 2, …, 12}, A = “a multiple of 3”, B = “even” and C = “greater than 8”. Is A independent of B? Is A independent of C?
A = {3, 6, 9, 12}. A ∩ B = {6, 12}: 2/12 = 1/6 = 1/3 × 1/2, independent. A ∩ C = {9, 12}: 2/12 = 1/6, but 1/3 × 1/3 = 1/9, dependent.
2
Hana and Yara each take one free throw, independently, with chances 0.5 and 0.4. Find P(both score), P(at least one scores), P(exactly one scores) and P(neither scores).
Both 0.2; neither 0.5 × 0.6 = 0.3; at least one 0.7; exactly one 0.5 × 0.6 + 0.5 × 0.4 = 0.5. Check: 0.2 + 0.5 + 0.3 = 1.
3
A pencil case holds 4 black and 2 red pens. Two are taken without replacement. Find P(both black), P(both red) and P(one of each).
Both black: 4/6 × 3/5 = 2/5. Both red: 2/6 × 1/5 = 1/15. One of each: 1 − 6/15 − 1/15 = 8/15.
4
A and B are independent with P(A) = 0.6 and P(B) = 0.5. Show that A and B′ are independent too.
P(A ∩ B′) = P(A) − P(A ∩ B) = 0.6 − 0.3 = 0.3, and P(A) × P(B′) = 0.6 × 0.5 = 0.3. The two are equal, so A and B′ are independent.
5
P(A) = 0.2 and P(B) = 0.3. Find P(A ∪ B) if A and B are mutually exclusive, and if they are independent.
Exclusive: 0.2 + 0.3 = 0.5. Independent: 0.2 + 0.3 − 0.06 = 0.44. The same two numbers give different unions, because the relations are different.