Transcript Finding all possible outcomes of two events

Combined Events
Sample Space Diagrams
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:
TT,
TH,
HT,
HH.
TH and HT are separate 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
HT
T
TH
TT
From the table we see that there are four possible outcomes one of which is two heads
so,
P(HH) =
1
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
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.
P(8) =
6
7
8
9
10
11
7
8
9
10
11
12
5
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
6
Fill in the rest of the cells in
the table.
3
First die
2
3
8
Colour in all the twos one colour,
the threes another colour etc.
4
5
6
9
What patterns do you notice in
the table?
Sample space diagrams
First die
Second die
+
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
1) What is the probability of getting a total
more than 5?
2) What is the probability of getting a total
less than 10?
3) What is the probability of getting a total
that is a square number?
4) What is the probability of getting an even
total?
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
7,H
8,T
Complete the table to show all the possible outcomes.
9
10
Combined events
Die
Coin
+
1)
2)
3)
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
How many outcomes are there?
What is the probability of getting a head and an even number?
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.
First die
Second die
+
1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
What is the probability of getting:
1) a total more than 4?
2) a total less than 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