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Probability Basics
Tree Diagrams, Sample Spaces, and Complementary Events
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Probability helps us measure how likely an event is to happen, from simple games to scientific experiments. Tree diagrams are a powerful way to organize all possible outcomes in a step by step process. They make it easier to see how events branch, count outcomes, and calculate probabilities correctly. Learning this skill builds a strong foundation for statistics, genetics, and decision making.
A tree diagram starts with one event and then splits into branches for each possible result. Each branch can split again if the experiment continues, such as flipping a coin twice or choosing items in sequence. The probability of one complete path is found by multiplying the probabilities along that path. Once all paths are shown, you can add the probabilities of paths that match the event you want.
Key Facts
- Probability of an event = number of favorable outcomes / total number of outcomes
- For one coin flip: P(H) = 1/2 and P(T) = 1/2
- For two independent events on one path: P(A and B) = P(A) x P(B)
- For two coin flips, the outcomes are HH, HT, TH, TT
- If outcomes do not overlap: P(A or B) = P(A) + P(B)
- The sum of probabilities of all final branches in a complete tree = 1
Vocabulary
- Outcome
- A single possible result of an experiment, such as getting heads on a coin flip.
- Event
- A set of one or more outcomes that match a condition you are interested in.
- Tree diagram
- A branching diagram that shows all possible outcomes of a sequence of events.
- Independent events
- Events are independent when the result of one does not change the probability of the other.
- Sample space
- The complete list of all possible outcomes of an experiment.
Common Mistakes to Avoid
- Forgetting to list every branch, which makes the sample space incomplete and leads to wrong probabilities. Check that each stage of the experiment splits into all possible outcomes.
- Adding probabilities along one path, which is wrong because sequential events on the same path must be multiplied. Use multiplication for and situations like H then T.
- Assuming every event has the same probability, which is not always true in multi step experiments. Count or calculate the probability of each final branch carefully.
- Mixing up outcomes and events, which causes confusion when counting favorable cases. An outcome is one result like HT, while an event can include several outcomes like getting exactly one head.
Practice Questions
- 1 A fair coin is flipped twice. Draw a tree diagram and find the probability of getting exactly one head.
- 2 A bag has 3 red marbles and 2 blue marbles. One marble is chosen, replaced, and then another is chosen. Use a tree diagram to find the probability of getting red then blue.
- 3 Explain why the probabilities at the ends of a complete tree diagram must add up to 1.