Probability and conditional probability
Use the whole sample space for regular probability, and shrink the denominator to the named subgroup for conditional probability.
Core Idea
Probability measures how likely an event is on a scale from 0 to 1. Conditional probability narrows the sample space to a specific subgroup before computing the likelihood.
Understanding
Core rule: probability is favorable outcomes divided by total outcomes.
- Regular probability: use the whole sample space.
- Conditional probability: shrink the denominator to the group named after "given" or "among".
- Two-way tables: row totals and column totals are the usual denominators.
- Trap: the numerator is still the favorable count; only the denominator changes.
Two extras show up often:
- Complement:
𝑃 ( n o t 𝐴 ) = 1 − 𝑃 ( 𝐴 ) - Independence:
𝑃 ( 𝐴 ∣ 𝐵 ) = 𝑃 ( 𝐴 )
Concept Guides
6Compute probability of an event from counts or a described sample space.
Probability is favorable outcomes divided by total outcomes.
Compute probability and conditional probability from a two-way table or area model.
Use the grand total for regular probability and the subgroup total for conditional probability.
Interpret conditional probability notation/results in context.
Read 𝑃(𝐴 ∣𝐵) as the probability of A among the B group.
Determine whether events are independent based on conditional probabilities (SAT).
Check independence by comparing 𝑃(𝐴 ∣𝐵) with 𝑃(𝐴), or by checking 𝑃(𝐴 and 𝐵) =𝑃(𝐴)𝑃(𝐵).
Apply complement and total probability ideas (sum of probabilities is 1).
Use the complement when counting the event directly is harder.
Interpret probability results and reasonableness in context.
A probability answer has to make sense in context.