Probability trees: independent events
Will it rain tomorrow, and will the school bus be late? Draw both on a probability tree, change their chances, and find the probability of both, neither, at least one and exactly one.
Core concept
For independent events, P(A and B) = P(A) × P(B); on a tree diagram you multiply along the branches and add the ends that give the outcome you want
Topic: Probability
Two events are independent when one happening does not change the chance of the other. Rain tomorrow (A) and the school bus being late (B) are taken as independent here. A probability: How likely something is to happen, from 0 (impossible) to 1 (certain). All key words tree shows them in two stages: first rain or no rain, then late or on time on each branch. The probabilities on the branches from one point add up to 1. The multiplication law: for independent events, P(A and B) = P(A) × P(B), so multiply along a path to get the chance of its end. The addition law for outcomes that cannot happen together: add the ends you want. So P(neither) = (1 − P(A))(1 − P(B)), P(at least one) = 1 − P(neither), and P(exactly one) adds the two mixed ends. When the first draw changes the second, as with two cards kept out of a pack, the events are dependent: see the lab on two cards in a row.
What you will learn
- Explain what independent events are.
- Draw a probability tree for two events.
- Use the multiplication law P(A and B) = P(A) × P(B).
- Add the ends of the tree for neither, at least one and exactly one.
Before you start
- Know that probabilities lie between 0 and 1 and P(not A) = 1 − P(A).
Materials
- A notebook to draw the tree
Equipment
- None
Procedure
- Record P(rain) = 0.3 and P(late) = 0.5.
- Make rain likely, P(rain) = 0.8, and record.
- Make the bus late less often, P(late) = 0.2, and record.
How this simulation is simplified
- Rain and the bus being late are independent of each other.