Independence of events is a key idea in probability because it tells us when knowing one outcome does not change the chance of another. If event A and event B are independent, the probability of B stays the same whether or not A has happened. This matters in statistics because it affects how we multiply probabilities, interpret data tables, and judge whether patterns are meaningful.
Many real situations, such as repeated coin flips, random sampling, and quality testing, depend on recognizing independence correctly.
Key Facts
- Events A and B are independent if P(A and B) = P(A)P(B).
- Equivalent test: A and B are independent if P(A | B) = P(A), as long as P(B) is not 0.
- Equivalent test: A and B are independent if P(B | A) = P(B), as long as P(A) is not 0.
- From a two-way table, compute P(A), P(B), and P(A and B), then compare P(A and B) with P(A)P(B).
- Worked example: If P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.20, then A and B are independent because 0.4 × 0.5 = 0.20.
- Mutually exclusive events with positive probabilities are not independent because P(A and B) = 0 but P(A)P(B) is greater than 0.
Vocabulary
- Event
- An event is a set of outcomes from a probability experiment, such as rolling an even number on a die.
- Independent events
- Independent events are events where the occurrence of one does not change the probability of the other.
- Dependent events
- Dependent events are events where the occurrence of one changes the probability of the other.
- Intersection
- The intersection of events A and B is the event that both A and B occur, written as A and B.
- Conditional probability
- Conditional probability is the probability that one event occurs given that another event has already occurred, written as P(A | B).
Common Mistakes to Avoid
- Assuming overlap means dependence is wrong because independent events can overlap as long as P(A and B) = P(A)P(B).
- Using P(A) + P(B) to test independence is wrong because independence is tested with multiplication, not addition.
- Calling mutually exclusive events independent is wrong when both events have positive probability because one event happening makes the other impossible.
- Checking only one raw count in a table is wrong because independence depends on probabilities or proportions, not just the size of one cell.
Practice Questions
- 1 A card is drawn from a standard 52-card deck. Let A be drawing a heart and B be drawing a queen. Find P(A), P(B), and P(A and B), then decide whether A and B are independent.
- 2 In a survey of 200 students, 80 play a sport, 50 play an instrument, and 20 do both. Use the multiplication test to decide whether playing a sport and playing an instrument are independent.
- 3 A student says that if two events are independent, they cannot happen at the same time. Explain why this statement is incorrect using the definition of independence.