Define probability. Start from equally likely outcomes, where a probability is one count divided by another, and compute probabilities by counting, including through the complement and in the birthday problem. Then decide when equally likely applies, and move to the general definition of probability and the properties that follow from it.
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Lesson 01 · The naive definition of probability
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The 7 lessons in this unit. Expand any one to see what it covers. Each unlocks when its own prerequisites are passed.
01
The naive definition of probability
11 questions
Learn how to tell when the outcomes of a random experiment are equally likely, and find the probability of an event by dividing the number of outcomes in it by the total number of outcomes.
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Equally likely outcomes
Probability as a ratio
02
Computing probabilities by counting
8 questions
Learn how to find a probability by counting when the outcomes are too many to list, building the favourable draws in stages and multiplying, and adding across the possible orders when draws are made with replacement.
Covers
Draws without replacement
Draws with replacement
03
The complement strategy
6 questions
Learn how to find the probability of "at least one" by working out the probability of "none" and subtracting it from 1, whether the draws are made with or without replacement.
Covers
The probability of at least one
04
When does "equally likely" apply?
9 questions
Learn when outcomes can be treated as equally likely, by symmetry or by design, and recognise when the naive definition of probability does not apply.
Covers
Justifying equally likely outcomes
When the naive definition of probability fails
05
The general definition of probability
13 questions
Learn the general definition of probability: a function on events that obeys two axioms, whether or not outcomes are equally likely. Then find an event's probability by adding the probabilities of its outcomes.
Covers
Probability as a function
The axioms of probability
Probability as a sum over outcomes
06
Properties of probability
13 questions
Learn three rules that follow from the axioms of probability and hold whether or not outcomes are equally likely: the complement rule, monotonicity and the general addition rule.
Covers
The complement rule
Monotonicity
The general addition rule
07
Collisions: the birthday problem
12 questions
Learn how to find the probability that no two people in a group share a birthday and the probability that someone shares one particular person's birthday, and see why a shared birthday becomes more likely than not with just 23 people.
Covers
The probability of no match
A match with a given person
Any match versus a match with a given person
This unit is part of our Essential Probability & Statistics for ML learning path, which contains 38 lessons. Every one of them is drawn below.
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7 lessons in this unit
27 prerequisite lessons across the unit, counting every step back to the start
4 other lessons in the learning path - after this unit, or alongside it
Builds on: Counting, Functions, Sample spaces and events, Set operations
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More in Probability and Statistics: Sample spaces and events, Counting