Probability Basics
Probability measures how likely something is on a scale from 0 (impossible) to 1 (certain). A handful of rules covers most everyday questions — and explains some famous surprises.
The basic idea
P(event) = favorable outcomes ÷ total outcomes
This works when every outcome is equally likely. A fair die has six equally likely faces, so P(rolling a 4) = 1/6 ≈ 0.167, or about 16.7%. Probabilities can be written as fractions, decimals or percentages.
Combining events
not A: P(not A) = 1 − P(A)
A and B (independent): P(A) × P(B)
A or B: P(A) + P(B) − P(A and B)
Draw one card from a standard 52-card deck
- P(heart) = 13/52 = 1/4
- P(king) = 4/52 = 1/13
- P(king of hearts) = 1/52
- P(heart or king) = 13/52 + 4/52 − 1/52 = 16/52
P(heart or king) = 4/13 ≈ 30.8%. The king of hearts is subtracted once so it is not counted twice.
“At least one” — use the complement
Questions about “at least one” are easiest from the other side: work out the chance of none, then subtract from 1.
At least one six in four rolls of a die
- P(no six in one roll) = 5/6
- P(no six in four rolls) = (5/6)⁴ ≈ 0.482
- P(at least one six) = 1 − 0.482
≈ 51.8% — slightly better than even.
The birthday problem
How many people do you need in a room for a better than even chance that two share a birthday? Most people guess far more; the answer is just 23. The chance that no two of k people share a birthday is (365/365) × (364/365) × … × ((365 − k + 1)/365); subtracting from 1 gives the table below (ignoring 29 February and assuming birthdays are spread evenly).
| People | Chance two share a birthday |
|---|---|
| 5 | 2.7% |
| 10 | 11.7% |
| 20 | 41.1% |
| 23 | 50.7% |
| 30 | 70.6% |
| 40 | 89.1% |
| 50 | 97% |
| 70 | 99.9% |
It feels surprising because there are many possible pairs: 23 people form 253 pairs, and any one of them can match.
Probability vs odds
Odds compare favorable to unfavorable outcomes. A probability of 1/6 is odds of 1 : 5 — one way for it to happen, five ways for it not to. Convert odds a : b to a probability with a ÷ (a + b).
Common mistakes
- Thinking past results change independent events. After five heads in a row, a fair coin still has a 1/2 chance of heads on the next toss. Coins and dice have no memory.
- Adding probabilities that overlap. “A or B” needs the overlap subtracted, or the total can exceed 1.
- Multiplying probabilities that are not independent. Drawing two aces without replacement is 4/52 × 3/51, not 4/52 × 4/52.
- Confusing small probabilities with impossibility. An event with a 1-in-1,000 chance per attempt becomes likely over thousands of attempts.