Probability and conditional information
By the end, you can
- Use complements, unions and intersections
- Distinguish independence from mutual exclusivity
- Calculate conditional probabilities using the right reference set
about 40 minutes for the lesson · practise at your own pace.
The central idea
What this chapter counts
The source outline covers addition and multiplication of probabilities, mutually exclusive and independent events, outcome tables, Venn and tree diagrams, counting with permutations and combinations, and conditional probabilities. Its three headline identities are the complement rule, the two-event addition rule, and the conditional probability formula.
The opening dice question below introduces probability through a gambling puzzle. The notes connect it to mid-seventeenth-century correspondence involving Blaise Pascal and Pierre de Fermat, then name quantum mechanics, gas kinetics, insurance risk and genetics as applications. The original dice photograph is decorative; the two mathematical events are retained in the activity.
Worked example
A bag contains three red and two blue balls. Drawing twice without replacement gives . Given that the first was red, the second-red probability is , not .
Mutually exclusive events with positive probabilities cannot be independent.
Check your understanding
Box A contains three red and one blue counter; box B contains one red and five blue counters. Choose a box with equal probability, then draw two counters from that box without replacement.
Find the probability of two reds and, given two reds, the probability that box A was chosen.
Compare with drawing two counters without replacement after pooling all ten counters.
Explain why the sampling mechanisms give different answers.
Let R mean two red draws. From A, ; from B it is zero. Hence and .
The pooled experiment has four red counters among ten, so its two-red probability is . Choosing a box equally does not select individual counters uniformly when the box sizes differ. It also makes both draws share the chosen box. The probability model must follow the actual selection mechanism and update the composition after the first draw.
Mastery learning objectives
At the end of this chapter, I should be able to:
Choose a practice question
Use the printed question-type map to choose a method to practise. Each link opens the complete question; the printed part references are retained.
| Skill | Lecture examples | Discussion | Review follow-up |
|---|---|---|---|
| Mutually exclusive events | 1, 3(ii) | 2(last part) | — |
| Independent events | 3(i) | 2(last part), 6(iv) | 1, 2, 4(iii)(iv), 6(i) |
| Table of outcomes | 4 | 1 | — |
| Venn diagrams | 5 | 2(i), 6 | 4, 5, 6(ii) |
| Permutations and combinations | 6, 9, 10 | 3, 4, 7 | 9, 10, 11, 14(a), 16 |
| Tree diagrams | 7, 8 | 5, 8 | 8, 12, 13, 14(b), 15 |
| Conditional probabilities | 2, 3, 4(iii)(iv), 7(iii), 8(iii), 10(c) | 1(v), 2(ii)(iii), 5(a)(iv) & (b), Q6(iii), Q8(ii) | 2, 3, 4(i)(ii), 6(i), 8(iii), 9(ii), 11(iii), 12(i), 13(i) & (ii)(b), 14(b)(ii), 15(ii), 16(ii) |
| Prior topic: geometric progression | 11, 12 | 7(iv) | — |
| Prior topic: applications of differentiation | — | 8(last part) | — |
| Future topic: binomial distribution | — | — | 12(iii) |
Check the part reference against the question. The conditional-probability reference Example 10(c) means 10(ii)(c).
The future-topic row prints Review 12(iii), but Review 12 has an initial unnumbered ask followed by (i) and (ii); its repeated-match calculation is (ii). The link opens that complete question.
Example 9 is listed under permutations and combinations although its printed solution uses direct replacement probabilities. Keep the source pointers while choosing the method from the actual question.
Practice questions
45 examples and practice questions, in 8 sections.
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