1
Counting – Learning Outcomes
 List all possible outcomes of an experiment or event.
 Use systematic listing.
 Use two-way tables.
 Use tree diagrams.
 Solve problems using the fundamental principle of
counting.
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List Outcomes
 When anything happens, we call it an event (or
sometimes an experiment).
 The specific way the event ends is called the outcome
of that event.
 The list of possible outcomes is called the sample space.
 e.g. rolling a die is an event.
 e.g. rolling a 4 on a die is an outcome.
 e.g. {1, 2, 3, 4, 5, 6} is the sample space of rolling a die.
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List Outcomes
 Most of the time, we see multiple related events.
 e.g. picking a main course AND a dessert at a
restaurant.
 e.g. choosing a car model AND a colour.
 e.g. rolling two dice.
 e.g. choosing which top, bottom, socks, AND shoes to
wear.
 e.g. seeing a film AND buying a drink.
 We need ways to write the sample spaces for these
kinds of events.
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Use Systematic Listing
 To make sure you don’t miss outcomes, it is best to list
them in some logical way.
 Try writing the sample spaces of these events
numerically or alphabetically:
 e.g. which suit is drawn from a standard deck of cards.
 e.g. which number is drawn from a standard deck of
cards.
 e.g. which card is drawn from a standard deck of cards.
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Use Systematic Listing
 If we have multiple events, try keeping all but one
outcome the same and just changing that one
outcome each time:
 e.g. if we have to choose between a red (R), blue (B) or
green (G) top, and a yellow (Y), orange (O) or purple (P)
bottom, list all the outfits that have a red top first, then all
the outfits with a blue top, then all the outfits with a
green top:
R+Y
R+O
R+P
B+Y
B+O
B+P
G+Y
G+O
G+P
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Use Systematic Listing
 List all the outcomes for a set menu if you must choose
one starter (fruit salad, mozzarella sticks, or vegetable
soup), one main course (steak, pasta, curry, or burger),
and one dessert (ice cream or trifle).
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Use Two-Way Tables
 If there are exactly two events, two-way tables work well
for listing outcomes.
 Write the outcomes from one event in the first column
and the outcomes from the other event in the first row.
 All the cells become the sample space for the
combination:
Red
Blue
Green
Yellow
R+Y
B+Y
G+Y
Orange
R+O
B+O
G+O
Purple
R+P
B+P
G+P
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Use Two-Way Tables
 Write out the possible meals you can have with a set
menu where you pick one starter (salad or soup) and
one main course (steak, chicken, or vegetarian
lasagne).
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Use Tree Diagrams
 For multiple events, tree diagrams show all the outcomes
easily.
 One event has branches for each outcome.
 Those outcomes have branches for each outcome of
the second event.
 Those outcomes have branches for each outcome of
the third event etc.
10
Use Tree Diagrams
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Use Tree Diagrams
 A game is played using the two spinners shown below.
 The first spinner has three segments labelled 2, 4, and 6.
The arrow has the same chance of stopping at each
number. The second spinner has six segments labelled A,
B, C, D, E, and F. The arrow has the same chance of
stopping at each letter. Two possible outcomes are (2,
A) and (6, D).
 Draw a tree diagram to show all the outcomes.
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Use Tree Diagrams
 When you reach a traffic light, there are three colours it
can be – red (R), amber (A), or green(G). Draw a tree
diagram to show the possible colours of two traffic lights
as you first arrive at them.