Total probability and Bayes' formula
Partition of the sample space, the total probability formula, Bayes' formula with its prior, likelihood and posterior probabilities, and the base rate fallacy with a test or filter that is rarely wrong.
- 27 cards
- 6 topics
- Free
The outline, topic by topic
Each topic is weighted the way the exam weighs it.
- Concepts
- Questions
- Sort and link
- The numbers
- Scenarios
- Understand
A few cards from the deck
Partition of the sample space: what is it?
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A family of events with non-zero probabilities, pairwise mutually exclusive, whose union is the whole sample space.
Each outcome belongs to exactly one event. “A and not A” is the simplest one; “server A, server B, server C” is another.
What condition is needed to apply the total probability formula with A₁, A₂ and A₃?
- That they form a partition of the sample space
- That they are pairwise independent
- That they all have the same probability
- That they all have probability below 1/2
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That they form a partition of the sample space
Each outcome must fall in exactly one Aᵢ: pairwise mutually exclusive, union equal to the sample space. Mutually exclusive events with non-zero probabilities are never independent, so that condition is even contradictory.
In Bayes' formula, P(A) is the … probability, P(B | A) the … and P(A | B) the … probability.
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In Bayes' formula, P(A) is the prior probability, P(B | A) the likelihood and P(A | B) the posterior probability.
We start from what we believed (prior), weigh it by how well the hypothesis explains the observation (likelihood), and get the updated belief.
Total probability formula: what is it?
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For a partition A₁, …, Aₙ, P(B) = P(B | A₁) × P(A₁) + … + P(B | Aₙ) × P(Aₙ): we add up the tree paths that lead to B.
It splits B according to the case of the partition where it occurs. It is the denominator of Bayes' formula.