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CIE A Level Maths: Probability & Statistics 1復習筆記2.1.3 Tree Diagrams

Category: A-level課程, 教材筆記, 福利干貨 Date: 2022年8月18日上午10:59

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?

3-1-3-fig1-tree-setup

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

3-1-3-fig2-pass-fail-tree

How do I solve probability problems involving tree diagrams?

Interpret questions in terms of?AND?and/or?OR?(See 1 Basic Probability)
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.

3-1-3-fig3-we-solution