Independent Events
In short: Independent events are a pair of happenings where hearing the result of one leaves the odds of the other untouched, so the odds of both come from multiplying.
GeometryProbabilityS-CP.A.2
What is Independent Events?
Two coin flips, a die roll and a spinner, or a free throw followed by another free throw are independent events, because in each case the first result tells you nothing about the second.
The test is a product rule: the pair \(A\) and \(B\) is independent exactly when \(P(A \text{ and } B) = P(A) \times P(B)\). Equivalently, \(P(A|B) = P(A)\). Drawing cards without putting them back breaks the rule, because each draw changes what is left in the deck.
How to spot it
- A story repeats the same chance mechanism, such as flips, spins, rolls or draws with replacement.
- The question says find the probability that both happen, or that a streak occurs.
- A table gives \(P(A)\), \(P(B)\) and \(P(A \text{ and } B)\) and asks whether the events are independent.
- A phrase such as does not affect, unrelated or regardless of is used.
The method
- Decide whether the first outcome could change the second chance; if not, treat the pair as independent events.
- Write each single probability in lowest terms or as a decimal.
- Multiply: \(P(A \text{ and } B) = P(A) \times P(B)\), extending to longer chains the same way.
- To test a claim of independence, compute the product and compare it with the given joint probability.
Independent Events: solved example
Problem
The chance of rain is 0.30 on Monday and 0.30 on Tuesday in a coastal town. Records show it rained on both days with probability 0.12. Are the two rainy days independent?
Solution
If they were independent, the chance of both would be the product \(0.30 \times 0.30 = 0.09\).
The recorded value is \(0.12\), which is not \(0.09\). Since \(P(A \text{ and } B) \neq P(A)P(B)\), the days are not independent: rain on Monday makes rain on Tuesday more likely.
Try it
Question 1
A spinner has numbers 1 to 5 and a fair coin is flipped. What is the probability of an odd number and tails?
Show the answer
The spinner and coin do not affect each other: \(\frac{3}{5} \times \frac{1}{2} = \frac{3}{10}\).
Question 2
Mia picks a heart-suit card from a full deck, puts it back, shuffles, and picks again. What is the probability she gets a heart both times?
Show the answer
Putting the card back keeps the picks unrelated: \(\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}\).
Independent Events vs. Mutually Exclusive Events
Mutually exclusive events cannot happen together, so \(P(A \text{ and } B) = 0\), and knowing one occurred tells you the other did not. Independent events can happen together, and their joint chance is the product of the two single chances. Two events with positive chances cannot be both exclusive and independent.
Keep going
Lessons
Worksheets and quizzes
- Conditional Probability and Independence – Worksheet
- Conditional Probability and Independence – Quiz
- Compound Events and Simulations – Worksheet
- Compound Events and Simulations – Quiz
Related words
- Conditional Probability
- Mutually Exclusive Events
- Probability
- Tree Diagram
- Compound Event
- Sample Space
Whole dictionary: A–Z · Geometry · Probability