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Statistics and Probability
Combined Events
Finding all possible outcomes of two events
Two coins are thrown. What is the probability of getting
two heads?
Before we can work out the probability of getting two heads
we need to work out the total number of equally likely
outcomes.
There are three ways to do this:
1) We can list them systematically.
Using H for heads and T for tails, the possible outcomes
are:
TH and HT are separate
TT, TH, HT, HH.
equally likely outcomes.
Finding all possible outcomes of two events
2) We can use a two-way table.
Second coin
First
coin
H
T
H
HH
TH
T
HT
TT
From the table we see that there are four possible outcomes
one of which is two heads so,
1
P(HH) =
4
Finding the sample space
A red dice and a blue dice are thrown and their scores
are added together.
What is the probability of getting a total of 8 from both
dice?
There are several ways to get a total of 8 by adding the
scores from two dice.
We could get a 2 and a 6, a 3 and a 5, a 4 and a 4,
a 5 and a 3, and a 6 and a 2.
To find the set of all possible outcomes, the sample
space, we can use a two-way table.
Finding the sample space
+
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
From the sample
space we can see
that there are 36
possible outcomes
when two dice are
thrown.
Five of these have
a total of 8.
5
P(8) =
36
Sample space diagrams
This table is another way of displaying all the outcomes from
throwing two dice and adding them together. It is called a
sample space diagram.
Second die
+
1
1
2
3
4
5
3
First die
2
3
8
4
Fill in the rest of the
cells in the table.
Colour in all the twos
one colour, the threes
another colour etc.
What patterns do you
notice in the table?
5
6
6
9
Sample space diagrams
First die
Second die
+
1
2
3
1
2
3
4
2
3
4
5
3
4
5
6
4
5
6
7
5
6
7
8
6
7
8
9
1) What is the probability of
4 5 6
getting a total more than 5?
5 6 7 2) What is the probability of
getting a total less than 10?
6 7 8
3) What is the probability of
7 8 9
getting a total that is a
square number?
8 9 10
4) What is the probability of
9 10 11
getting an even total?
10 11 12 5) What is the probability of
getting an odd total?
6) What is the probability of
getting a total less than 13?
Combined events with coins and dice
A coin and a ten-sided die are thrown and the outcomes
recorded in the two-way table below.
Dice
Coin
+
H
T
1
2
2,H
3
4
5
6
7
8
9
10
7,H
8,T
Complete the table to show all the possible outcomes.
Combined events
Die
Coin
+
1
2
3
4
5
6
7
8
9
10
H
1,H 2,H 3,H 4,H 5,H 6,H 7,H 8,H 9,H 10,H
T
1,T 2,T 3,T 4,T 5,T 6,T 7,T 8,T 9,T 10,T
1) How many outcomes are there?
2) What is the probability of getting a head and an even
number?
3) What is the probability of getting a tail and a square
number?
Other combined events
Two four-sided dice are thrown and the numbers added
together.
Construct a sample space diagram to show all the outcomes.
What is the probability of getting:
First die
Second die
+
1
2
3
4
1) a total more than 4?
1
2
3
4
5
2) a total less than 8?
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
3) a prime number total?
4) a total that is at least 3?
5) a total of 4 or 5?
6) the same number on both dice?
7) a lower number on the first dice?
Calculating the number of outcomes