Conditional probability
Answers are for parents and teachers.
1
Two dice are thrown. Find P(the sum is 10 given that at least one die shows 6).
At least one 6: 11 pairs. Sum 10 among them: (4, 6) and (6, 4). So the probability is 2/11.
2
A library recorded 100 borrowers (table). Find P(fiction | adult), P(adult | fiction) and P(child and non-fiction).
| Child | Adult | Total | |
|---|---|---|---|
| Fiction | 28 | 24 | 52 |
| Non-fiction | 12 | 36 | 48 |
| Total | 40 | 60 | 100 |
P(fiction | adult) = 24/60 = 0.4. P(adult | fiction) = 24/52 = 6/13. P(child and non-fiction) = 12/100 = 0.12.
3
P(A) = 0.7, P(B | A) = 0.4 and P(B) = 0.5. Find P(A ∩ B), P(A | B) and P(A ∪ B).
P(A ∩ B) = 0.4 × 0.7 = 0.28; P(A | B) = 0.28 ÷ 0.5 = 0.56; P(A ∪ B) = 0.7 + 0.5 − 0.28 = 0.92.
4
A basket holds 3 oranges and 7 apples. Two pieces of fruit are taken without replacement. Draw the tree and find P(two oranges), P(orange then apple) and P(two apples).
P(OO) = 3/10 × 2/9 = 1/15; P(OA) = 3/10 × 7/9 = 7/30; P(AA) = 7/10 × 6/9 = 7/15. Check: 1/15 + 7/30 + 7/30 + 7/15 = 1.
5
In a school P(girl) = 0.5, P(choir) = 0.2 and P(girl and choir) = 0.15. Find P(choir | girl) and P(girl | choir), and say which is larger and why.
P(choir | girl) = 0.15 ÷ 0.5 = 0.3 and P(girl | choir) = 0.15 ÷ 0.2 = 0.75. The second is larger because the choir is a smaller group than the girls, so the same overlap is a bigger share of it.