Probability Calculator
Calculate the probability of single and multiple events occurring
Probability Formulas
Understanding Probability
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.
Types of Probability Calculations
Single Event
P(A) = favorable/total. The basic probability of one event occurring.
Union (OR)
P(A or B) for when either event occurring counts as success.
Intersection (AND)
P(A and B) for when both events must occur together.
Complement (NOT)
P(not A) = 1 - P(A). The probability an event doesn't occur.
Common Probability Examples
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 |
Key Probability Concepts
Independent Events
Events are independent when one doesn't affect the other's probability. Coin flips are independent - previous results don't change future probabilities.
Mutually Exclusive
Events are mutually exclusive when they can't occur together. Rolling a 3 and a 5 on one die throw are mutually exclusive.
Conditional Probability
P(A|B) is the probability of A given that B occurred. It's calculated as P(A and B) / P(B).
Law of Large Numbers
As trials increase, observed frequency approaches theoretical probability. This is why casinos always win long-term.
Frequently Asked Questions
How do I convert probability to odds?
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.
What's the gambler's fallacy?
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.
How do I calculate 'at least one' probability?
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.
What's expected value?
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.
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