Probability
Basic probability, tree diagrams, Venn diagrams, conditional probability
Basic Probability
Probability measures how likely an event is to occur. It is always a value between 0 (impossible) and 1 (certain).
P(event) = Number of favourable outcomes ÷ Total number of equally likely outcomes
Events that cannot both occur at the same time. P(A or B) = P(A) + P(B).
Events where the outcome of one does not affect the outcome of the other. P(A and B) = P(A) × P(B).
P(A does not happen) = 1 − P(A). The complement of an event A is written A' or Ā.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) For mutually exclusive events: P(A ∪ B) = P(A) + P(B)
Tree Diagrams
Tree diagrams show all possible outcomes of two or more events and their probabilities.
- Multiply along branches to find the probability of a sequence of events
- Add the probabilities of different branches that give the same outcome
- All probabilities on branches from the same point must sum to 1
- The probabilities of all final outcomes must sum to 1
A bag contains 3 red and 2 blue balls. Two balls are drawn without replacement. Find P(both red).
Without replacement: the denominator decreases for the second pick. With replacement: the probabilities are the same for both picks.
Venn Diagrams
Venn diagrams use overlapping circles to show the relationships between sets and their probabilities.
Everything in A or B or both. P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Everything in both A and B. The overlapping region.
Everything NOT in A. P(A') = 1 − P(A).
Fill in the Venn diagram from the inside out: start with the intersection, then fill in the remaining parts of each circle, then the outside region. All regions must sum to 1 (or the total frequency).
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred.
P(A | B) = P(A ∩ B) ÷ P(B) Read as: 'the probability of A given B'
P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2. Find P(A | B).
Conditional probability questions often use the phrase 'given that'. When drawing a tree diagram, conditional probabilities appear on the second set of branches.