What is probability?
Probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain). A probability of 0.5 means the event is expected half the time. When all outcomes are equally likely, the probability of an event is the number of favourable outcomes divided by the total number of outcomes.
Theoretical vs. experimental probability
- Theoretical probability is worked out by reasoning: a fair die has 6 equally likely faces, so P(6) = 1/6.
- Experimental probability (or relative frequency) is measured: if you roll 600 times and get 94 sixes, the experimental probability is 94/600 ≈ 0.157.
The two are rarely exactly equal, especially for small numbers of trials. The difference is not a mistake — it is natural random variation.
The law of large numbers
As the number of trials increases, the experimental probability tends to get closer and closer to the theoretical probability. In the graph, the line jumps around wildly at first and then settles near the dashed line. Note that the count can still drift away from the expected count — only the proportion becomes stable. Roughly speaking, quadrupling the number of trials halves the typical size of the random error in the proportion.
Combining events: two dice
Two dice have 6 × 6 = 36 equally likely combinations, but the sums are not equally likely. A sum of 7 can be made in six ways, while a sum of 2 can only be made as 1 + 1. That is why the bar chart for two dice forms a triangle peaking at 7. Because the two dice do not affect each other, they are independent, and the probability of a specific pair (such as a 3 then a 5) is 1/6 × 1/6 = 1/36.
Formulas
P(event) = favourable outcomes / total equally likely outcomesExperimental probability = number of times the event occurred / number of trialsExpected count = P(event) × number of trialsIndependent events: P(A and B) = P(A) × P(B)Key terms
- Trial
- One repetition of an experiment, such as one flip or one roll.
- Outcome and event
- An outcome is a single result; an event is a set of outcomes we are interested in (for example “rolling an even number”).
- Sample space
- The set of all possible outcomes.
- Independent events
- Events where one does not change the probability of the other.
- Biased
- Not fair — some outcomes are more likely than they should be.
Real-world applications
- Genetics: Punnett squares give the probability that offspring inherit a trait, such as a 1 in 4 chance for a recessive condition when both parents are carriers.
- Medicine: clinical trials use large samples so that random variation does not hide or fake a treatment’s effect.
- Weather forecasts: a “70% chance of rain” means that in similar conditions it rained about 70% of the time.
- Insurance and engineering: companies use probabilities of rare events, such as floods or component failures, to plan and set prices.
- Simulation: scientists run millions of random trials on computers (Monte Carlo methods) to study systems too complex to calculate directly.
Common misconceptions
- The gambler’s fallacy: “After five heads, tails is due.” Each flip is independent.
- “Results should match theory exactly.” Small samples vary a lot; that variation is expected.
- “All sums of two dice are equally likely.” Sums near 7 have more combinations.
Key takeaways
- Probability ranges from 0 to 1 and can be calculated or measured.
- Experimental results vary, especially with few trials.
- With many trials, experimental probability approaches the theoretical value.
- Independent events have no memory: past results do not change future chances.
Model note: the simulator uses your browser’s pseudo-random number generator, which is suitable for teaching but not for cryptography.