AQA A Level Maths: Statistics复习笔记3.1.3 Tree Diagrams
Tree Diagrams
What is a tree diagram?
- A tree diagram is used to
- show the (combined) outcomes of more than one event that happen one after the other
- help calculate probabilities when AND and/or OR’s are involved
- Tree diagrams are mostly used when there are only two mutually exclusive outcomes of interest
- e.g. “Rolling a 6 on a die” and “Not rolling a 6 on a die”
- More than three outcomes per event can be shown on a tree diagram but they soon become difficult to draw and so lose their effectiveness
- Tree diagrams are very helpful when probabilities for a second event change depending on the first event
How do I draw and label a tree diagram?
- In the second experiment, P(B) may be different on the top set of branches than the bottom set
e.g. This is most commonly seen in drawing one item at random, not replacing it, then drawing another
- Sometimes a second branch may not be needed following a first event
- e.g. In aiming to pass a test (experiment) the event fail on the first attempt would require a second attempt but the event pass on the first attempt would not
How do I solve probability problems involving tree diagrams?
- Interpret questions in terms of AND and/or OR (See 1 Basic Probability)
- Draw, or complete a given, tree diagram
- When you are confident with tree diagrams you can just pull out the (final) outcome(s) you need to answer a question rather than routinely list all of them
Worked Example
A contestant on a game show has three attempts to hit a target in a shooting game.
They have a maximum of three attempts to hit the target in order to win the star prize – a speedboat. If they do not hit the target within three attempts, they do not win anything.
The probability of them hitting the target first time is 0.2. With each successive attempt the probability of them failing to hit the target is halved.
Find the probability that a contestant wins the star prize of a speedboat.
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