Probability
Mutually exclusive outcomes
If you roll a 1 on a dice, you cannot also have rolled a 3. Outcomes that cannot happen together are mutually exclusive — and if they cover everything, their probabilities add to 1.
Outcomes that cannot happen together
Two outcomes are MUTUALLY EXCLUSIVE when they cannot both happen at the same time. Rolling a 1 and rolling a 3 are mutually exclusive — one roll cannot be both. Rain and strong wind are NOT mutually exclusive. A day can easily be both wet and windy.
Equally likely outcomes
When every outcome is EQUALLY LIKELY, you do not need an experiment. You can calculate. On a fair dice, 1, 2, 3, 4, 5 and 6 are all equally likely. But for a football match, winning 1–0 and winning 6–0 are NOT equally likely — so you cannot just count outcomes there.
Theoretical probability
When outcomes are equally likely, probability = number of ways it CAN happen ÷ total number of outcomes Ten cards show 3, 5, 5, 8, 5, 9, 8, 5, 3, 8. One is taken without looking. P(5) = 4/10 = 2/5 = 40% P(8) = 3/10 = 30% P(1) = 0/10 = 0 — impossible
Adding to 1
If a set of mutually exclusive outcomes covers everything that could happen, their probabilities add to 1. BANANA has 6 letters: 1 B, 3 A, 2 N. P(B) = 1/6, P(A) = 3/6, P(N) = 2/6 1/6 + 3/6 + 2/6 = 6/6 = 1 ✓ This gives a shortcut: P(not A) = 1 − P(A).
Marcus's mistake
Marcus says: 'My team can win, draw or lose. Winning is one of three outcomes, so P(win) = 1/3.' That is wrong. Dividing by the number of outcomes only works when the outcomes are EQUALLY LIKELY — and win, draw and lose are not. Three outcomes does not mean one-third each.
Worked examples
Ten cards: 3, 5, 5, 8, 5, 9, 8, 5, 3, 8 — P(5)?
- Count the 5s: there are 4.
- There are 10 cards altogether.
- 4/10 simplifies to 2/5.
Answer: 2/5, or 40%
P(8) from the same ten cards
- Count the 8s: there are 3.
- 3 out of 10.
Answer: 3/10, or 30%
P(1) from the same ten cards
- No card shows a 1.
- 0 out of 10 is impossible.
Answer: 0
P(an even number) on a fair dice
- The even faces are 2, 4 and 6 — that is 3 outcomes.
- There are 6 faces altogether.
- 3/6 simplifies to 1/2.
Answer: 1/2
BANANA — do the probabilities add to 1?
- P(B) = 1/6, P(A) = 3/6, P(N) = 2/6.
- 1/6 + 3/6 + 2/6 = 6/6.
Answer: Yes — they add to 1, because they cover every card
Six boys and four girls — P(not a girl's name)?
- P(girl) = 4/10 = 2/5.
- Use P(not A) = 1 − P(A).
- 1 − 2/5 = 3/5.
Answer: 3/5
Where you see this in real life
- A raffle where each ticket is equally likely to win. Your chance is just your tickets over the total sold.
- Quality checks on a production line assume each item is equally likely to be picked, so the sample represents the batch.
Key words
- Mutually exclusive
- two outcomes that cannot both happen
- Equally likely
- outcomes with the same chance of happening
- Theoretical probability
- a probability worked out by counting, not by experiment
- Outcome
- one possible result
- Fair
- having every outcome equally likely; also called unbiased