Probability
Probability: rules, trees and relative frequency
Probability is measured on a scale from 0 to 1. For equally likely outcomes, P(event) = favourable outcomes ÷ total outcomes, and P(not A) = 1 − P(A). A sample space diagram or two-way table lists all outcomes for combined events.
Two events are mutually exclusive if they cannot both happen, so P(A or B) = P(A) + P(B). They are independent if one does not change the probability of the other, so P(A and B) = P(A) × P(B). Tree diagrams handle multi-stage experiments: multiply along the branches and add between the branches that satisfy the event.
Trap: with 'without replacement', the totals on the second set of branches change - always update both the numerator and the denominator. For 'at least one', use the complement: 1 − P(none). Do not confuse mutually exclusive with independent; they are different ideas.
Relative frequency = number of successes ÷ number of trials, and it estimates the theoretical probability. The estimate generally improves as the number of trials increases, but it gives no guarantee for any fixed number of trials.
Worked example. A bag holds 3 red and 2 blue counters; two are taken without replacement. P(both red) = 3/5 × 2/4 = 6/20 = 3/10. P(at least one red) = 1 − P(both blue) = 1 − (2/5 × 1/4) = 1 − 1/10 = 9/10.