Probability, trees and conditional reasoning · Extension
| English | 中文 | Pinyin |
|---|---|---|
| conditional probability/kənˈdɪʃənl ˌprɒbəˈbɪlɪti/ | 条件概率 | tiáo jiàn gài lǜ |
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.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines conditional probability?
The probability of an event after restricting the sample space to a stated condition.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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.
Without replacement, P(RR)=3/5×2/4=0.3.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Mutually exclusive events with positive probabilities must be independent. This claim is false. Explain which definition or assumption it violates.
Find P(A given B) if P(A∩B)=0.12 and P(B)=0.3.
P(A|B)=P(A∩B)/P(B)=0.12/0.3=0.4.
Mutually exclusive events with positive probabilities must be independent.
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.
Interpret a new situation
- 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.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the probability of one red and one blue in that bag.
Add the two orders: 3/5×2/4+2/5×3/4=0.6.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 9260 · Extension · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The probability of an event after restricting the sample space to a stated condition. Choose the relationship, show the method, check its assumptions and interpret the result.