Coin flip guide
Coin Flip Odds: Ratios and Percentages
Probability compares successful outcomes with all outcomes. Odds compare successful outcomes with unsuccessful ones.
Probability versus odds
For probability p, odds in favor are p:(1−p), and odds against are (1−p):p. Heads on a fair flip has probability 1/2 and odds of 1:1. The phrase ‘one in two’ describes the probability, not the odds ratio.
Work through an example
Exactly two heads in three flips occurs in HHT, HTH, and THH. Three of eight sequences succeed and five fail. Probability is 37.5%; odds in favor are 3:5 and odds against are 5:3.
Specify which run counts
HHH in a specified three-flip block has probability 1/8. Allowing HHH or TTT doubles it to 1/4. A run somewhere in a longer session is a separate event with more possible starting positions.
Read the table carefully
The table uses independent fair flips and specified blocks. These are mathematical ratios, not offered payouts. Use the probability guide to learn how to count outcomes, or enter a particular event in the calculator.
| Event | Probability | Odds in favor |
|---|---|---|
| Heads on one flip | 50% | 1:1 |
| HH in two flips | 25% | 1:3 |
| HHH in three flips | 12.5% | 1:7 |
| HHH or TTT in three flips | 25% | 1:3 |
| Exactly two heads in three flips | 37.5% | 3:5 |
Sources and further reading
Worked examples use the independent fair-coin model unless a section explicitly discusses physical tossing.