The probability scale
Introduced| English | Meaning |
|---|---|
| probability/ˌprɒbəˈbɪlɪti/ | A numerical measure of how likely an event is, ranging from zero to one. |
| outcomes/ˈaʊtkʌmz/ | A possible or actual result of an event, process or trial. |
| complement/ˈkɒmplɪmənt/ | The set of elements in a specified universal set that do not belong to the given set. |
How likely is that?
- The chance of being struck by lightning in a year is about $0.000001$. The chance of flipping heads is $0.5$. The chance of the sun rising tomorrow is $1$.
- Probability 概率 puts every event on a single scale from impossible ($0$) to certain ($1$).
The probability scale
- $\text{P}(\text{event})$ measures how likely it is, from $0$ to $1$.
- $0$ = impossible, $\dfrac{1}{2}$ = even chance, $1$ = certain.
- Can be written as a fraction, decimal, or percentage.
A fair spinner: eight identical sectors, so each number is an equally likely outcome
The probability scale
Every probability sits between 0 (impossible) and 1 (certain). Slide the marker along the 0–1 scale.
What is the probability of an impossible event?
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Impossible events have probability 0; certain events have probability 1.
Write the probability of an "even chance" as a decimal.
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An even chance is ½ = 0.5.
A certain event has probability 100.
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A certain event has probability 1 (or 100%), not 100. Probability is between 0 and 1.
Calculating probability
- For equally likely outcomes 结果:
A bag has $3$ red and $5$ blue counters. $\text{P}(\text{red}) = \dfrac{3}{8}$.
Dice: the probability scale runs from impossible (0) to certain (1)
A bag has 3 red and 5 blue counters. P(red) = a/8. What is a?
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3 red out of 8 total: P(red) = 3/8.
For equally likely outcomes, P(event) = favourable outcomes / ______ outcomes.
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P = favourable / total, when all outcomes are equally likely.
The complement 补集
- The probability an event does not happen:
Use the complement when it's easier. Finding P(at least one six in 10 rolls) directly is hard. Finding P(no sixes) and subtracting from 1 is much simpler.
Probability runs from $0$ (impossible) to $1$ (certain), with $\tfrac12$ an even chance
A bag has 3 red and 5 blue counters. P(not red) = a/8. What is a?
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P(red) = 3/8, so P(not red) = 1 − 3/8 = 5/8; a = 5.
Fun fact
- The study of probability began in the 1600s when French gamblers asked mathematicians to solve problems about dice games. Pascal and Fermat's letters founded the whole subject.
The probability scale is a number line from $0$ to $1$ — every event sits somewhere on it.
You've got it
- probability runs $0$ (impossible) to $1$ (certain); $\dfrac{1}{2}$ is even chance
- $\text{P} = \dfrac{\text{favourable}}{\text{total}}$ for equally likely outcomes
- complement: $\text{P}(\text{not } A) = 1 - \text{P}(A)$