To reason about uncertainty, first separate the possible outcomes from the events you care about.
A coin toss produces heads or tails. We cannot know the next outcome in advance, but we do know what can happen.
Probability studies situations in which individual outcomes are uncertain but the structure of possible outcomes is known.
Key question
What can happen, which outcomes matter, and how likely are they?
What are trials, sample spaces, and events?
Trial
One repeatable process whose outcome is uncertain
Sample space
The set of all possible outcomes
Event
A collection of outcomes we care about
Probability
A number from 0 to 1 describing how likely an event is
One roll of a die
Trial: roll the die once
Sample space: {1, 2, 3, 4, 5, 6}
Event A: an even result = {2, 4, 6}
How is probability calculated?
Probability of event A
outcomes in A ÷ all possible outcomes
Three of the six die outcomes are even, so the probability is 3/6 = 0.5.
| Probability | Meaning |
|---|---|
| 0 | Impossible |
| 0.5 | An even chance |
| 1 | Certain |
How do unions and intersections differ?
A union contains outcomes in A or B. An intersection contains outcomes that belong to both A and B.
| Expression | Notation | Example |
|---|---|---|
| A or B | A ∪ B | even or at least 4 |
| A and B | A ∩ B | even and at least 4 |
| not A | Aᶜ | not even |
Do not count the overlap twice
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Theoretical and empirical probability
A fair coin has theoretical heads probability 0.5, but ten actual tosses might produce seven heads.
The observed proportion is an empirical probability. It can vary widely in small samples and tends to settle toward the theoretical probability as the number of trials grows.
A common mistake
Five heads in a row do not make tails more likely on the next independent toss.
Change the number of trials
Change the success probability and trial count to see how much the observed proportion can move.
Trials
20
Expected probability
0.50
Observed successes
7
Observed proportion
0.35
Key lesson
Probability describes a long-run pattern across repeated trials, not a guarantee about the next outcome.