Xác suất từ bảng dữ liệu và một nhóm giới hạn
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| probability/ˌprɒbəˈbɪlɪti/ | A numerical measure of how likely an event is, ranging from zero to one. |
| complement/ˈkɒmplɪmənt/ | Two angles that add up to 90 degrees, or the event that something does not happen. |
| independent/ˌɪndɪˈpendənt/ | Describing quantities, events or choices that do not depend on one another in the relationship being considered. |
A survey table
- A survey of 200 students records whether they live on campus and whether they own a bike. "If a student who lives on campus is chosen at random, what is the probability 概率 they own a bike?"
- The key word is the condition: "who lives on campus" changes the total you divide by.
Probability from counts
- $P(\text{event}) = \dfrac{\text{favourable outcomes}}{\text{total outcomes}}$ when outcomes are equally likely.
- The complement 补事件 "not A" has probability $1 - P(A)$: if $P(\text{rain}) = 0.3$, then $P(\text{no rain}) = 0.7$.
The on-campus row gives P(bike | on campus) = 45 / 120
The probability that a bus is late is 0.15. What is the probability that it is not late?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$1 - 0.15 = 0.85$.
Conditional probability: restrict the group
- "Given that / of those who / if a … is chosen" restricts the denominator to that group.
- Table: on campus 120 (bike 45, no bike 75); off campus 80 (bike 50, no bike 30). $P(\text{bike} \mid \text{on campus}) = \dfrac{45}{120} = 0.375$.
Same table: $P(\text{on campus} \mid \text{bike}) = \dfrac{45}{95}$. Swapping the condition changes the denominator from 120 to 95.
Which number goes in the denominator?
Choose the correct whole for a probability.
Of 150 people, 60 drink coffee; 24 of the coffee drinkers also drink tea. If a coffee drinker is chosen at random, what is the probability that they drink tea?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The condition (coffee drinker) restricts the denominator to 60.
In the survey table, 50 of the 80 off-campus students own a bike. What is $P(\text{bike} \mid \text{off campus})$ as a decimal?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$50 \div 80 = 0.625$.
Independent 独立的 versus disjoint
- Disjoint events cannot both happen (rolling a 2 and a 5 on one roll). Independent events do not affect each other's probability (two separate coin flips).
- Disjoint events with nonzero probabilities are never independent: if one happens, the other cannot.
Two disjoint events with nonzero probabilities cannot be independent.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. If one happens the other cannot, so P(A and B) = 0, which is not P(A)P(B) when both are nonzero.
Traps
- Wrong denominator: using the grand total (200) when the question restricts to a group.
- Reversing the condition: $P(A \mid B)$ is not $P(B \mid A)$.
The probability that a student owns a bike given they live on campus always equals the probability that a student lives on campus given they own a bike.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The denominators are different groups.
Recap
- Probability = favourable ÷ total; complement = $1 - P$.
- A condition restricts the denominator to that group; disjoint ≠ independent.
- complement: Two angles that add up to 90 degrees, or the event that something does not happen.