One flip
A coin has two outcomes: H (heads) and T (tails). A fair coin means each one is just as likely:
- P(heads) = 1/2
- P(tails) = 1/2
The coin has no memory
Every flip is independent: a fresh start that isn’t affected by earlier flips. Even if a coin lands on heads 5 times in a row, the chance of heads next time is still 1/2. Work out each flip on its own.
More than one flip: list every outcome
| Flips | All the possible outcomes | How many |
|---|---|---|
| 1 | H, T | 2 |
| 2 | HH, HT, TH, TT | 4 |
| 3 | HHH, HHT, HTH, HTT, THH, THT, TTH, TTT | 8 |
Each extra flip doubles the number of outcomes: 2, 4, 8, 16…
How to list 3 flips without missing any: take the 4 outcomes for 2 flips (HH, HT, TH, TT). Put an H in front of each one (HHH, HHT, HTH, HTT), then a T in front of each one (THH, THT, TTH, TTT). That makes all 8.
Then underline the outcomes the question asks for, and write underlined ÷ total.
- P(heads every time in 2 flips) = 1/4. In 3 flips: 1/8. In 4 flips: 1/16.
- P(exactly one head in 2 flips): HT or TH = 2 out of 4 = 1/2
- P(at least one tail in 2 flips): everything except HH = 3/4
How many would you expect?
Flip a coin 10 times and you’d expect about 5 heads (half of 10). You won’t always get exactly 5, but the more you flip, the closer you get to half.
Watch out! HT and TH are different outcomes: heads first then tails isn’t the same as tails first then heads. Count them both.
Top tip: for “at least one”, it’s often quicker to find what’s left: P(at least one tail) = 1 − P(no tails at all).
A coin is flipped 3 times. Tap every outcome with exactly two heads.
🪙