Coins: list every result
You toss a big coin and a small coin together. On each coin, head and tail have the same chance. What is the chance that the big coin shows a head?
- The big coin shows head (H) or tail (T).
- The small coin shows H or T after each one.
- There are 4 results. All four are equally likely.
- A head on the big coin: HH, HT. That is 2 of 4. P = 1/2.
The big coin shows a head in 2 of the 4 results. So the chance is 1/2.
The rule
A fair coin treats head and tail alike. Head and tail are equally likely. For one coin, P(H) = 1/2.
With more coins, one rule keeps the count right.
List every result in order. HT and TH are different.
Write the big coin's result first, then the small coin's.
- In HT, the big coin shows a head.
- In TH, the small coin shows a head.
A new coin adds H or T to every result. So the list doubles. Two coins give 4 results. Three coins give 8.
Three coins
| Heads | Results | How many |
|---|---|---|
| 0 | TTT | 1 |
| 1 | HTT, THT, TTH | 3 |
| 2 | HHT, HTH, THH | 3 |
| 3 | HHH | 1 |
The groups hold 1, 3, 3 and 1 results. So the groups are not equally likely.
Example. What is P(at least two heads)? At least two heads means two or three heads. That is 3 + 1 = 4 results. P = 4/8 = 1/2.
Second example. At least one head means any result except TTT. That is "not TTT". P = 1 − 1/8 = 7/8.
Now try it. Slide the coins. Pick a question.
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