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Tree diagrams

Tree diagrams are useful for organising and visualising the different possible outcomes of a sequence of events. For each possible outcome of the first event, we draw a line where we write down the probability of that outcome and the state of the world if that outcome happened. Then, for each possible outcome of the second event we do the same thing.

Below is an example of a simple tree diagram, showing the possible outcomes of rolling a 6-sided die.

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Note that each outcome (the numbers 1 to 6) is shown at the end of a line; and that the probability of each outcome (all in this case) is shown shown on a line. The probabilities have to add up to 1 in order to cover all of the possible outcomes. In the examples below, we will see how to draw tree diagrams with multiple events and how to compute probabilities using the diagrams.

Earlier in this chapter you learned about dependent and independent events. Tree diagrams are very helpful for analysing dependent events. A tree diagram allows you to show how each possible outcome of one event affects the probabilities of the other events.

Tree diagrams are not so useful for independent events since we can just multiply the probabilities of separate events to get the probability of the combined event. Remember that for independent events:
So if you already know that events are independent, it is usually easier to solve a problem without using tree diagrams. But if you are uncertain about whether events are independent or if you know that they are not, you should use a tree diagram.

Exercise 1: Tree diagrams

You roll a die twice and add up the dots to get a score. Draw a tree diagram to represent this experiment. What is the probability that your score is a multiple of 5?

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The tree diagram for the experiment is shown above. To save space, probabilities were not indicated on the branches of the tree, but every branch has a probability of . The multiples of 5 are underlined. Since the probability of each of the underlined outcomes in and since there are 7 outcomes that are multiples of 5, the probability of getting a multiple of 5 is .

What is the probability of throwing at least one five in four rolls of a regular 6-sided die? Hint: do not show all possible outcomes of each roll of the die. We are interested in whether the outcome is 5 or not 5 only.

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The outcomes that lead to at least one 5 in four rolls of the die are marked on the tree diagram above. Summing the probabilities along all the branches gives

You flip one coin 4 times.

  1. What is the probability of getting exactly 3 heads?

  2. What is the probability of getting at least 3 heads?

  1. There are possible outcomes for 4 coin tosses. There are 4 outcomes that contain exactly 3 heads, namely , , , . Therefore the probability of getting exactly 3 heads is

  2. Getting at least 3 heads is the same as getting either exactly 3 heads or exactly 4 heads. We have already seen that there are 4 ways to get exactly 3 heads. There is 1 way of getting exactly 4 heads, namely the outcome . Hence there are 5 ways of getting at least 3 heads and the probability of this event is .

You flip 4 different coins at the same time.

  1. What is the probability of getting exactly 3 heads?

  2. What is the probability of getting at least 3 heads?

  1. The mathematics of this problem is exactly the same as the previous problem since it does not matter whether we flip 4 different coins at the same time or the same coin 4 different times. The correct answer is .

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