DPST1012 Chap.11 Probability, Counting and the Binomial Theorem
Probability, Counting and the Binomial Theorem
Probability as set language
Topic 8 runs in Week 10 and reuses the set language of the first week. The syllabus lists basic probability, Venn diagrams, the additive and complement rules, random variables, successive events and the multiplicative rule, the test for independence, dependent events and conditional probability, counting, the binomial theorem and binomial probability.
An event is a set of outcomes, so union, intersection and complement keep their meanings and the addition rule carries straight over.
Successive and dependent events
When events happen in stages, a tree diagram organises the work: multiply along a path to get the probability of that sequence, and add the paths that make up the event wanted. The second-stage numbers on a tree are conditional probabilities.
Two events are independent when the probability of both equals the product of their separate probabilities; otherwise knowing one changes the chance of the other. Drawing without replacement is the standard example of dependence.
Counting and the binomial
Counting supplies the numerators of many probabilities. Permutations count arrangements where order matters and combinations count selections where it does not.
The same combination numbers are the coefficients in the binomial theorem for expanding a power of a + b, and they reappear in binomial probability, which gives the chance of exactly k successes in n independent trials with the same probability of success on each. 'At least one' questions are almost always quickest through the complement.
What this chapter covers
- 01
Events as sets and Venn diagrams
- 02
Addition and complement rules
- 03
Tree diagrams and the multiplicative rule
- 04
Conditional probability
- 05
Testing for independence
- 06
Permutations and combinations
- 07
The binomial theorem
- 08
Binomial probability
Worked example · free
Conditional probability from a two-stage tree
- 1P(both red) = 5/8 × 4/7 = 20/56 = 5/14.
- 1P(red then blue) = 5/8 × 3/7 = 15/56 and P(blue then red) = 3/8 × 5/7 = 15/56.
- 1P(one of each) = 15/56 + 15/56 = 30/56 = 15/28.
- 1P(second red | first red) = 4/7, because 4 red remain among 7 tokens.
Key terms
- Conditional Probability
- The probability of an event given that another has occurred, P(B | A) = P(A ∩ B) / P(A).
- Independent Events
- Events for which the probability of both occurring equals the product of their separate probabilities.
- Combination
- A selection of objects in which order does not matter, counted by n choose r.
- Permutation
- An arrangement of objects in which order matters, counted by n factorial over (n − r) factorial.
- Binomial Probability
- The probability of exactly k successes in n independent trials, each with the same success probability p.
Probability, Counting and the Binomial Theorem FAQ
What is the difference between mutually exclusive and independent?
Mutually exclusive events cannot happen together, so their intersection is empty. Independent events can happen together, and one occurring does not change the chance of the other. Events with non-zero probabilities cannot be both.
When should I use a tree diagram instead of a formula?
Whenever events happen in stages and the later probabilities depend on what happened earlier, such as drawing without replacement. The tree shows every path, so no case is missed.
Why use the complement for 'at least one' questions?
Because 'at least one' covers many cases, while its opposite, 'none', is a single case. One minus the probability of none is usually a single multiplication instead of a long sum.
When can I use the binomial formula?
When there is a fixed number of trials, each with two outcomes, the same success probability every time, and independence between trials. Repeated dice rolls qualify; drawing cards without replacement does not.
How can I tell whether order matters in a counting question?
Ask whether swapping two chosen items gives a different outcome. Arranging people in a queue or assigning first and second prizes depends on order, so use permutations; picking a committee or a hand of cards does not, so use combinations.
Exam move
Write each event in set notation before calculating, keep answers as fractions until the last line, and check every tree by confirming that the final probabilities sum to 1. Rewrite one worded question a day as set notation before solving it.
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