Probability and Events
Statistics through experiments · 1 of 16 · CC BY-SA 4.0
You can know the chances and still be surprised by what happens. Let’s give that a try.
The wheel below has two colors. Its pointer can land in any direction, with each direction equally likely. Half the wheel is blue, so blue has half the chance.
Before you spin it ten times, take a guess: how many blue results will you get? Keep your guess in mind, then press Spin 10 times.
Every direction has the same chance.
Changing the chance starts a fresh record.
No spins yet. Both colors are possible.
The wheel makes no promises about ten spins
Perhaps you got five blues. Perhaps you didn’t. Both are perfectly ordinary results. A half-blue wheel gives blue a 50% chance on each spin; it doesn’t keep a little ledger and stop at five.
Try a few more batches. The percentage you observe can move toward 50%, then away again. Over many independent spins it tends to settle near the wheel’s chance, but it needn’t get closer with every new spin.
A moment to think
A name for the chance
Probability is a number from 0 to 1 that describes how likely an event is under a model. Here our model is especially simple: every direction on the wheel is equally likely.
Choose 25% blue. That is a probability of 0.25, or one chance in four. At 0, blue cannot happen on this wheel. At 1, every spin is blue. Try those settings too; the ends help make the scale feel familiar.
One spin is a trial. The color it lands on is an outcome. An event is an outcome, or a group of outcomes, that we’re interested in. Our event is “lands on blue.”
The wheel’s probability and your observed percentage answer different questions: What are the chances? and What happened this time? Keeping those apart will help throughout the book.
Does a coin remember?
Imagine a fair coin has just landed heads four times: H, H, H, H. It’s tempting to give tails the next turn. But a fair coin has no turn-taking rule.
When flips are independent, knowing earlier results does not change the chance of the next one. Our spinner works that way too. A streak leaves its colored areas exactly as they were.
A moment to think
A little shorthand to take with you
You have chosen an event and assigned it a chance. There is a compact way to write those two things together. Let A stand for the event, and p for its probability.
You can now write this
Choose a piece to read it in words.
Read P as “the probability of.” It asks for a chance, expressed as a number from 0 to 1.
A quarter of the spinner is blue → the chance of blue is 0.25.
For the half-blue wheel, you can write P(blue) = 0.5. Read it aloud: “The probability of blue is one half.” The notation carries the idea you have already been using.
A moment to think
Words you met
| Term | Meaning |
|---|---|
| Trial | One performance of the experiment: a spin, a flip, or a draw. |
| Outcome | The result of a trial. |
| Event | The outcome or group of outcomes we care about. |
| Probability | A chance described by a number from 0 to 1. |
| Observed frequency | How often the event actually occurred. Dividing its count by the number of trials gives its relative frequency. |
| Independence | Knowing one result does not change the probability of another. |
Neighbors
- Next, we’ll flip a handful of coins at once and see what happens when we repeat the whole experiment. For the formal rules, visit Probability.
- For the mathematical foundations, the probability textbook’s Discrete Probability chapter develops sample spaces and events. Its Law of Large Numbers chapter explains why repeated trials help frequencies settle toward probabilities.
Written by June Kim. The conversational approach was inspired by Danielle Navarro’s Learning Statistics with R (CC BY-SA 4.0). The prose, experiments, and questions here were created for this book. This chapter’s text and illustrations are also shared under CC BY-SA 4.0.