Addition theorem of probability sheet5-mark answer frameTwo dice are rolled. Find the probability that the sum is 8 or both dice show the same number.
How to lay out the answer. Follow the frame, then write it in your own words.
- 1 · Write what is given and what is to be found n(S) = 36. A: the sum is 8; B: a doublet. To find: P(A ∪ B).
- 2 · Write the theorem Addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- 3 · List event A
- A = {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}, n(A) = 5
- P(A) = 5/36
- 4 · List event B
- B = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, n(B) = 6
- P(B) = 6/36
- 5 · Find the overlap
- A ∩ B = {(4, 4)}, so P(A ∩ B) = 1/36
- 6 · Substitute
- P(A ∪ B) = 5/36 + 6/36 − 1/36
- = 10/36 = 5/18
- 7 · Check and box the answer P(A ∪ B) = 5/18. Check by counting: A ∪ B has 5 + 6 − 1 = 10 outcomes out of 36.
Scoring tip: Always look for the overlap before adding. Here (4, 4) is in both events; forgetting it gives 11/36, which is wrong.
