Probability, trees and conditional reasoning
Введено| English | Значение |
|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | условная вероятность |
What changes after the first draw?
- A bag has 3 red and 2 blue counters. Taking two without replacement changes the chance of the second colour.
- This lesson studies conditional probability 条件概率: The probability of an event after restricting the sample space to a stated condition.
The method
- Multiply along a tree branch and add disjoint branches.
- With replacement, the composition stays fixed.
- Conditional probability is P(A given B)=P(A∩B)/P(B), for P(B)>0.
Two disjoint events A and B both have positive probability. Can they be independent?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Disjoint means A and B cannot happen together, so P(B given A)=0≠P(B). Independence requires that knowing one does not change the other - disjoint events with positive probability always fail that.
A worked case
Without replacement, P(two red)=3/5×2/4=3/10. P(one of each)=3/5×2/4+2/5×3/4=3/5. If P(A∩B)=0.12 and P(B)=0.3, P(A given B)=0.4.
Probability, trees and conditional reasoning
Multiply along a tree branch and add disjoint branches
Compare the model with the worked case and explain one change.
Find the probability of two red from 3 red and 2 blue without replacement.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Without replacement, P(RR)=3/5×2/4=0.3.
Where it goes wrong
- Mutually exclusive means no overlap; independent means that knowing one event does not change the other's probability.
- Two disjoint events with positive probability are not independent.
Not true: Mutually exclusive events with positive probabilities must be independent. Mutually exclusive events with positive probabilities are dependent: if one happens, the other cannot, so P(A and B)=0≠P(A)P(B).
Find P(A given B) if P(A∩B)=0.12 and P(B)=0.3.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
P(A|B)=P(A∩B)/P(B)=0.12/0.3=0.4.
Mutually exclusive events with positive probabilities are dependent.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
True. Mutually exclusive events with positive probabilities are dependent: if one happens, the other cannot, so P(A and B)=0≠P(A)P(B).
Applying it
- A two-way table makes the restricted denominator visible.
- Before using a product P(A)P(B), justify independence from the context or the supplied information.
Find the probability of one red and one blue in that bag.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Add the two orders: 3/5×2/4+2/5×3/4=0.6.
Match each tree rule to its use.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Branches multiply because later probabilities apply only after earlier outcomes; disjoint routes to one outcome add.
In your course
- AQA A-level Mathematics (7357), specification section M.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Conditional probability: The probability of an event after restricting the sample space to a stated condition.
- Multiply along a tree branch and add disjoint branches.
- Watch for: mutually exclusive means no overlap; independent means that knowing one event does not change the other's probability.