Thinking Skills: Critical Thinking and Problem Solving

6.3 Probability, tree diagrams and decision trees 241

Commentary

A way of solving this problem using a tree

diagram is shown below. At each stage (i.e. as

each coin is drawn from the pocket) the

branches of the tree lead to the possibilities –

in this case only the withdrawal of a 5¢ or 10¢

coin – and the numbers beside the branches

show the probability of each outcome. After

three coins are withdrawn, the totals of all

possible combinations of coin value may be

calculated (by adding coin values along the

branches) and the probability of that

combination obtained (by multiplying the

probabilities along the branches). After

making the calculations, you may check

whether you are correct as the sum of the

probabilities should be 1.

The problem may now be solved. Reading

from the top, combinations 2, 3 and 5 lead to a

sum of 20¢. The sum of the probabilities for

these three combinations is^15 +^15 +^15 =^35 , i.e.

a 60% chance.

you lose with 2 of these (blue 1 and 2) and you
win with 4 (blue 3, 4, 5 or 6).
So the number of ways of winning is 12 + 4
= 16 out of 36. (The number of ways of losing is
18 + 2 = 20 out of 36.) So the probability of
winning is^1636 =^49.
The examples below are more complex and
are more likely to relate to Advanced Level and
university entrance examinations.

Tree diagrams
Tree diagrams can be of help especially in
probability problems that are not absolutely
straightforward. They enable probabilities for
every combination of events to be evaluated,
and allow probabilities to be divided between all
possible circumstances. They also give the
advantage that, as all probabilities are calculated,
we can check that the sum of them is 1.

I have six coins in my pocket: four of 5¢ and

two of 10¢. If I take three coins out of my

pocket at random, what are the chances of

the total being 20¢?

Activity

3

5

2

(^235)
1
3 4
5
1
5
1
4
1
4
3
4
1
3 ××= 0
1
5
0
1
2
15¢ 3 ××
20¢
20¢
25¢
20¢
25¢
25¢
30¢
5¢
5¢
1
(^0) 10¢
10¢
10¢
3
5
1
2
1
5
2
3 ××
3
5
1
2
1
5
2
3 ××
2
5
3
4
1
5
1
3 ××
4
5
3
4
1
5
1
3 ××
4
5
1
4
1
15
1
3 ××
1
5
1
1
1
15
2
3 ××
= = = = = =
(^2514) = 151
3
4
1
2
1
2
5¢
10¢
5¢
10¢
5¢
10¢
5¢
5¢
10¢