Combinations in probability
- Once counting is solid, probability is just favourable ways over total ways.
- The numerator and denominator must count the same kind of thing.
- For selections, use combinations. For seatings or arrangements, use arrangements.
Good selections over all selections
. Count both parts the same way.
Worked example
First rung: one boy and one girl
Choose 2 children from 6 boys and 4 girls. Find the probability of one boy and one girl.
- Favourable selections:.
- Total selections:.
Answer
Worked example
Exactly 5 soup tins
There are 8 soup tins and 7 non-soup tins. Seven tins are selected. Find the probability of exactly 5 soup tins.
- Favourable selections:.
- Total selections:.
- Divide: .
Answer
Worked example
Repeated letters in a selection probability
Four letters are selected from the 9 letters in ANDROMEDA. Find the probability of at least one D and exactly one A. (9709/52/O/N/23 Q7(c))
- Scenario AD plus two other letters:.
- Scenario ADD plus one other letter:.
- Favourable selections: .
- Total selections: .
Answer
| Question says | Denominator |
|---|---|
| sample of r, selected at random | |
| random arrangement / random seating | |
| at least one | often use 1 − P(none) |
| more than / at least / at most | split into legal cases and add |
Your turn— tap to reveal the worked answer (selection probability)
Choose 2 children from 6 boys and 4 girls. Find P(one boy and one girl).
- Favourable selections:.
- Total selections: .
Answer
Common mistake
If the numerator counts selections, the denominator must also count selections. Mixing on top with underneath is usually wrong.