Introduction to Probability
Introduced| English |
|---|
| sample space/ˈsæmpl speɪs/ |
| outcome/ˈaʊtkʌm/ |
| complement rule/ˈkɒmplɪmənt ruːl/ |
| equally likely/ˈiːkwəli ˈlaɪkli/ |
| Venn diagram/ven ˈdaɪəɡræm/ |
All the ways it could go
- The sample space 样本空间 is the set of all possible outcomes of a chance process.
- Each individual result is an outcome 结果; an event is a set of outcomes.
- Rolling a die: sample space $\{1,2,3,4,5,6\}$; the event "even" $= \{2,4,6\}$.
- Listing the sample space is the first step in finding a probability.
The basic rules
- Every probability satisfies $0 \le P(A) \le 1$ (never negative, never above $1$).
- The probabilities of all outcomes in the sample space sum to $1$.
- Complement rule 补集法则: $P(A^c) = 1 - P(A)$ — "not $A$" is whatever's left.
- These rules constrain every valid probability model.
Equally likely outcomes
- When outcomes are equally likely 等可能, probability is just counting.
- $P(A) = \dfrac{\text{number of outcomes in } A}{\text{total number of outcomes}}$.
- $P(\text{even on a die}) = 3/6 = 0.5$.
- This "favorable over total" rule works only when outcomes are equally likely.
Picturing events
- A Venn diagram 韦恩图 draws events as overlapping circles inside the sample space.
- The overlap is the intersection (both events); the whole shaded region is the union (either).
- A two-way table does the same job with counts in rows and columns.
- Both make "and," "or," and "not" easy to see and compute.
The counting rule $P(A)=\frac{\text{favorable}}{\text{total}}$ only works when outcomes are equally likely. For a biased coin or unequal outcomes, you must use the given probabilities, not just count. And always check your model obeys the basics: every $P$ in $[0,1]$, and the total over the sample space equal to $1$.
Draw one card from a standard $52$-card deck.
- Sample space: the $52$ cards (equally likely). Event "heart" has $13$ outcomes.
- $P(\text{heart}) = 13/52 = 0.25$.
- Complement: $P(\text{not a heart}) = 1 - 0.25 = 0.75$.
The sample space is all possible outcomes; every model has $0 \le P(A) \le 1$ with the total equal to $1$, and the complement rule $P(A^c) = 1 - P(A)$. For equally likely outcomes, $P(A) = \frac{\text{favorable}}{\text{total}}$. Represent events with a Venn diagram or two-way table.
Events as overlapping circles
A Venn diagram shows unions, intersections, and complements.
Rolling a fair die, what is P(even)? Give a decimal.
Even = {2,4,6}, so 3/6 = 0.5.
If P(A) = 0.25, what is P(not A) by the complement rule?
P(A^c) = 1 − 0.25 = 0.75.
The rule P(A) = favorable/total works only when the outcomes are...
Counting works only for equally likely outcomes.
A valid probability can be 1.4.
Every probability must satisfy 0 ≤ P ≤ 1.
The set of all possible outcomes of a chance process is the sample ___.
The sample space lists every possible outcome.