Page 4 - Pra U STPM 2022 Penggal 3 - Maths (T)

Mathematics Semester 3 STPM Chapter 2 Probability

2.1 Counting Techniques

Addition principle of counting

Let A , A , ..., A be disjoint events with n , n , …, n possible outcomes, respectively. Then the total number
k
k
2
1
2
1
of outcomes for the event “A or A or ... or A ” is n + n + … + n .
1 2 k 1 2 k
Suppose that we want to buy a fruit from one of two stalls A and A . Suppose also that those stalls have 10
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and 15 different types of fruits, respectively. How many types of fruits are there altogether to choose from?
2 Choosing one from given types of fruits from either stalls is called an event and the choices for either
event are called the outcomes of the event. Thus the event “selecting one from stall A ”, for example, has 10
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outcomes. Essentially, the addition principle says that if we want to count the number of ways either one
case could happen or another case could happen, then we should add the number of ways each individual
case could happen. Thus, we can choose one of 10 types of fruits from stall A or one of 15 types of fruits
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from stall A , there are altogether 10 + 15 = 25 types of fruits to choose from.
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Note that the events must be disjoint, that is they must not have common outcomes for this principle to
be applicable.
Example 1

Suppose there are 4 different flavours of noodle dishes and 7 different ingredients of fry rice dishes. How
many selections does a customer have?

Solution: An event is “selecting a dish of either kind”.
There are 4 outcomes for the noodle event and 7 outcomes for the rice event.
According to the addition principle, there are 4 + 7 = 11 possible selections.

Multiplication principle of counting

Let A , A , ..., A be events with n , n , ..., n possible outcomes, respectively. Then the total number of
1
k
k
2
2
1
outcomes for the sequence of these k events is n × n × … × n .
k
1
2
If we are buying a cup of ice cream that comes in a choice of three flavours from vanilla, chocolate or
mango, and two sizes either small cup or large cup, how many different types of ice creams can be ordered?
We have three choices for the flavours, for each choice of flavour; there are two choices of sizes. Selecting one
of three choices is called an event, and a specific size is called the outcome of the event. The multiplication
principle tells us that if we want to count the number of ways that one case could happen and another case
could happen, then we should multiply the number of ways that each individual case could happen. Thus,
we could order 3 × 2 = 6 different types of ice cream.

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02 STPM Math(T) T3.indd 72 28/10/2021 10:21 AM