M4.6.4 Chapter 4

Tree diagrams

A bag has 3 red and 2 blue counters. I take one, look, put it back, and take another. What is the chance of one of each colour?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A 6/25, from 3/5 × 2/5.
You picked A. Read on. At the end, the chapter comes back to your guess.
B 12/25, because red then blue and blue then red both count.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 1/2, because there are two colours.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Draw a tree diagram for two events
Find probabilities by multiplying along the branches

A table lists the outcomes of two events well. When there are more events, or the chances are not equal, a tree diagram is better. It follows the events in order, one stage at a time.

Start with a coin tossed twice. The first toss is H or T, and each has chance 1/2. From each of those, the second toss is H or T again.

Draw it first

Draw a tree for two tosses of a coin

1 Draw the first toss: two branches, H and T, each marked 1/2. Then draw the second toss from the end of each.
2 At each end, write the outcome and its chance. Multiply the branches on the way: 1/2 × 1/2 = 1/4.

Two rules make a tree work. The branches that leave one point add up to 1, because one of them must happen. And along a path, you multiply, because each path is "this and then that".

Worked example

A fair coin is tossed twice. What is the probability of two heads?

The path for two heads is H then H.
Each branch has probability 1/2.
Multiply along the path: 1/2 × 1/2 = 1/4.

Answer: 1/4

See it three ways

With things

Toss a fair coin twice and write H or T for each toss. Do it a few times. Two heads comes up in 1 of every 4 goes, about.

As a picture

In words

List: the outcomes are HH, HT, TH and TT, and one of the 4 is HH. Table: one cell of 4 is HH. Tree: the path H then H has probability 1/2 times 1/2. All three ways give 1/4.

In symbols

P(HH) = 1/4 = 1/2 × 1/2.

Now the bag. It holds 3 red and 2 blue counters, so P(red) = 3/5 and P(blue) = 2/5. We put the counter back each time, so the second pick has the same chances as the first.

The bag holds 5 counters. Each counter is put back after it is taken.

The two hatched ends are one of each colour. Their chances are 6/25 each.

To find the chance of an event, find every end where it happens. Work out each end by multiplying. Then add those ends, because the ends cannot happen together.

Worked example

A counter is taken from the bag of 3 red and 2 blue, then put back, and a second is taken. What is the probability of one of each colour?

Red then blue: 3/5 × 2/5 = 6/25.
Blue then red: 2/5 × 3/5 = 6/25.
These are two different paths, so add them: 6/25 + 6/25 = 12/25.

Answer: 12/25

A good check: all the ends together must add to 1, because something must happen. Here 9/25 + 6/25 + 6/25 + 4/25 = 25/25 = 1.

A tree can have more stages. Toss a coin three times. Each stage doubles the number of ends, so there are 8 ends. Each end has chance 1/2 × 1/2 × 1/2 = 1/8.

Three tosses. The three hatched ends have exactly two heads.

Worked example

A fair coin is tossed three times. What is the probability of exactly two heads?

The paths with exactly two heads are HHT, HTH and THH. That is 3 paths.
Each path has probability 1/2 × 1/2 × 1/2 = 1/8.
Add the three paths: 3 × 1/8 = 3/8.

Answer: 3/8

Try it

Test the tree with a bag

You need: 3 white stones and 2 dark stones, a cloth bag or a cup, a pencil and paper.

Put the stones in the bag. Pick one without looking, write its colour, and put it back.
Pick again, write the colour, and put it back.
Do this 50 times. Count the times you got one white and one dark.
The tree says 12/25 of the pairs, so you expect about 24 of 50.

What you will see: Your count is near 24, but not exactly 24. The tree gives the chance. The experiment only estimates it.

If you cannot try it, this is what happens.

A wrong idea many people have

To find the chance of a path, add the branch chances.

Why it feels right: We add chances for "or", so it seems we should add them here too.

The test that breaks it: Try red then red with chances 3/5 and 3/5. Adding gives 6/5, which is more than 1. No chance can be more than 1.

What is true: A path is "this and then that". Multiply along it. Add only where different paths end in the same event.

Key words

tree diagram: a diagram of branches that shows the outcomes of events in order, with a chance on each branch.
branch: one line of a tree diagram, marked with a probability.

Back to your guess

The answer is B: 12/25, because red then blue and blue then red both count.

One of each can happen two ways: red then blue, and blue then red. Each way has chance 3/5 × 2/5 = 6/25. Add the two ways and you get 12/25. A tree diagram shows both ways at once.

Keep it

Copy this into your exercise book.

A tree shows events in order. Branches from one point add to 1. Multiply along a path. Add the ends that make the event. All ends add to 1.

Draw: A tree for two tosses of a coin, with 1/2 on each branch and 1/4 at each end.