Calculate the probability of single and multiple events occurring
Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0 means the event is impossible, while 1 means it's certain. Most real-world events fall somewhere in between.
The basic probability formula divides favorable outcomes by total possible outcomes. For a fair coin flip, the probability of heads is 1/2 = 0.5 (one favorable outcome out of two possible). For a standard die, rolling a 6 has probability 1/6 ≈ 0.167.
Probability theory forms the foundation of statistics, risk assessment, game theory, and machine learning. Understanding probability helps make better decisions under uncertainty.
P(A) = favorable/total. The basic probability of one event occurring.
P(A or B) for when either event occurring counts as success.
P(A and B) for when both events must occur together.
P(not A) = 1 - P(A). The probability an event doesn't occur.
Reference table for everyday probability scenarios:
| Event | Probability | Percentage | Odds |
|---|---|---|---|
| Coin flip (heads) | 0.5 | 50% | 1:1 |
| Die roll (specific) | 0.167 | 16.7% | 1:5 |
| Card (specific) | 0.019 | 1.9% | 1:51 |
| Card (suit) | 0.25 | 25% | 1:3 |
| Two heads in row | 0.25 | 25% | 1:3 |
| Lottery (6/49) | 0.0000001 | 0.00001% | 1:13.9M |
Events are independent when one doesn't affect the other's probability. Coin flips are independent - previous results don't change future probabilities.
Events are mutually exclusive when they can't occur together. Rolling a 3 and a 5 on one die throw are mutually exclusive.
P(A|B) is the probability of A given that B occurred. It's calculated as P(A and B) / P(B).
As trials increase, observed frequency approaches theoretical probability. This is why casinos always win long-term.
Odds = P/(1-P). For probability 0.25, odds are 0.25/0.75 = 1:3 or '1 to 3'. This means 1 success for every 3 failures expected.
The mistaken belief that past events affect future independent probabilities. If a coin lands heads 10 times, the next flip is still 50/50 - coins have no memory.
Use the complement: P(at least one) = 1 - P(none). For rolling at least one 6 in three rolls: 1 - (5/6)³ ≈ 0.42.
Expected value = Σ(outcome × probability). It's the average result if you repeated the experiment many times. For a fair die: (1+2+3+4+5+6)/6 = 3.5.
You want the probability of flipping two heads in a row with a fair coin, and separately the probability that two fair six-sided dice sum to 7.
ResultP(two heads) = 0.25; P(sum = 7) = 1/6 ≈ 0.1667
Coin flips are independent, so P(H and H) = P(H) × P(H) = 0.5 × 0.5 = 0.25. For two dice, there are 6 × 6 = 36 equally likely ordered outcomes. The pairs summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — six of them — giving 6/36 = 1/6 ≈ 0.1667.
Independent events do not influence each other: P(A and B) = P(A) × P(B). Two coin flips or two separate dice rolls are independent. Dependent events change each other's probability — for example, drawing two cards without replacement. For dependent events you use the multiplication rule P(A and B) = P(A) × P(B|A), where the second probability is conditional on the first outcome.
Conditional probability P(A|B) is the probability of event A given that event B has already occurred. It is calculated as P(A|B) = P(A and B) / P(B), provided P(B) > 0. Conditioning on B effectively shrinks the sample space to just the outcomes where B happens, then asks what fraction of those also include A.
Bayes' theorem reverses a conditional probability: P(A|B) = P(B|A) × P(A) / P(B). Use it when you know how likely the evidence is under each hypothesis but want to update the probability of the hypothesis after seeing the evidence. It underpins medical test interpretation, spam filtering, and most modern statistical inference.
Use the inclusion–exclusion principle: P(A or B) = P(A) + P(B) − P(A and B). The subtraction prevents double-counting outcomes that are in both A and B. When A and B are mutually exclusive, P(A and B) = 0, so the formula simplifies to P(A) + P(B).
Probability is the fraction of outcomes that are favorable: p = favorable / total, always between 0 and 1. Odds compare favorable to unfavorable outcomes as a ratio: odds = p / (1 − p), often written like 1:3 or 3 to 1. A probability of 0.25 corresponds to odds of 1:3 (one win for every three losses).
The gambler's fallacy assumes independent past events affect future ones (a fair coin is not 'due' for tails after several heads). The base rate fallacy ignores prior probability when interpreting test results. The conjunction fallacy treats P(A and B) as larger than P(A). And confusing P(A|B) with P(B|A) — the prosecutor's fallacy — causes serious errors in medical and legal reasoning.
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