WJEC MATHEMATICS
INTERMEDIATE
STATISTICS AND PROBABILITY
PROBABILITY AND TREE
DIAGRAMS
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Contents
All Probabilities are Between 0 and 1
Probabilities Add up to 1
Listing All Outcomes
Expected Probability
The AND / OR Rule
Tree Diagrams
Probability from Venn Diagrams
Credits
Probability scale
https://sites.google.com/a/egrps.org/murphys-math/probability-1
WJEC Question bank
http://www.wjec.co.uk/question-bank/question-search.html
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All Probabilities are between 0 and 1
Probabilities are always between 0 and 1. The higher the probability of
something, the more likely it is to happen
 A probability of 0 means it NEVER WILL happen
 A probability of 1 means it DEFINITELY WILL happen
Probability formula
Probability = Number of ways for something to happen
Total number of possible results
Example
Calculate the probability of selecting a vowel from the tiles below
Probability =
3
Probabilities Add up to 1
There are two key facts we need to know:
1. If only one result can happen at a time, then all the probabilities will add
up to one
2. Since something must either happen or not happen;
The probability it happens + The probability it doesn't happen = 1
Example 1
Example 2
The probability John is late for work is 0.26. What is the probability he is not
late for work
From the second key fact
1 - 0.26 = 0.74
Exercise S4
1. Complete the following tables
Colour
Probability
Red
0.52
Transport
Probability
Bike
0.24
Card
Probability
Card 1
0.45
Green
0.3
Car
0.41
Card 2
0.15
Train
Blue
Walk
0.16
Card 3
Plane
0.14
Other
0.03
Card 4
0.09
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2.
One of the following shapes are chosen at random
What is the probability of that shape being a triangle?
What is the probability of that shape being a square?
What is the probability of that shape being a circle?
What is the probability of that shape not being a circle?
3. Jamie selects a number between (and including) 1 and 25.
a) What is the probability that his selected number is a multiple of 6?
b) What is the probability that his selected number is a multiple of 5?
c) What is the probability that his selected number is a square number?
d) What is the probability that his selected number is prime?
e) What is the probability that his selected number is not a 1 digit
number?
4. Alice selects a letter a random from the word BANANA.
a) What is the probability that her selected letter is the letter N?
b) What is the probability that her selected letter is a vowel?
c) What is the probability that her selected letter is a T?
5. The probability Jasmine drives to work is 0.73. Calculate the probability
that she does not drive to work
6. The probability of it being sunny tomorrow is
. What is the probability
that it will not be sunny?
7. When rolling a fair 20 sided die, what is the probability of not rolling a factor
of 24?
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Listing All Outcomes
Listing all outcomes, is just listing all the things that could happen. Often, we
are asked to create a sample space diagram.
A sample space diagram is a good way to show all the possible outcomes if
there are two activities going on (e.g. two coins being thrown, two dice being
thrown, or two spinners).
Example
The following two spinners are spun and the numbers on both are multiplied
together. Create a sample space diagram to show all possible outcomes.
Spinner 1
1
3
5
2
6
10
2
4
12
20
4
6
18
30
6
8
24
40
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Spinner 2
The number of values in the table (highlighted) gives us the total number of
possible results. This is very useful for using the probability formula. For
example, you may be asked to calculate the probability of your score being 6
There are 12 items in the table so
there are 12 possible outcomes
There are 2 sixes in the table. i.e.
there are 2 ways of getting a 6
Probability =
6
Expected Frequency
Expected times something will happen = probability x number of trials
Example 2
The probability of winning a game is . If a player plays the game 180 times,
how many would you expect them to win?
Using the above formula:
Expected number of wins =
x 180 = 60
Exercise S5
1.
7
2.
3.
8
4.
5.
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Relative Frequency
Some probabilities we know (an example would be the probability of flipping a
coin and it landing on heads)
If we don't know the probability, we can calculate an estimate of it through
repeated experiment. In this case, instead of using the word 'Probability' we
use "Relative Frequency"
The following table shows results of 100 rolls of an untested die.
Score
1
2
3
4
5
6
Frequency
3
20
50
7
15
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Relative Frequency =
(a) What is the relative frequency of obtaining a 3
(b) What is the probability of scoring 5 or more
(c) If die is rolled 600 times, how many times would you expect
to get a 1
From above, it shows you get
3 1s in 100 rolls, so will get 3 x 6 = 18 1s in 100 throws
Note: From above, it seems as if the dice is unfair as you would
expect approximately 17 for each value (100 6)
MORE ROLLS (TRIALS) WOULD MAKE THE RESULTS MORE
RELIABLE
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Exam Questions S7
1.
11
2.
12
And / Or Rule
AND
OR
x
+
If you are ask to find the probability of event A and event B, you multiply the
probabilities together
If you are asked to find the probability of event A or event B, you add the
probabilities together
Example 1
A bag contains 5 red balls, 4 yellow balls, and 3 green balls. One ball is
randomly selected from the bag. Find the probability that the selected ball is
red or yellow.
P(red) =
P(yellow) =
P(red or yellow) =
Example 2
The probability that Jane wears a dress to work is 0.3. The probability that
she walks to work is 0.2. Find the probability that Jane wears a dress and
she walked
P(dress) = 0.12
P(walk) = 0.2
P(dress and walk) = 0.3 x 0.2 = 0.6
Example 3
The probability Jane wears a hat is 0.3. The probability she wears a hat and
eats a burger is 0.12. Find the probabiliy she eats a burger
0.3 x P(Burger) = 0.12
P(Burger)= 0.4
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Tree Diagrams
A tree diagram is a way of seeing all the possible probability 'routes' for two
(or more) events. A game consists of selecting a counter from a bag
(containing three red counters and seven blue) twice.
Important
Each set of lines
that meet at the
same point
MUST add to 1
Important
When travelling along branches, you MULTIPLY
Question 1: Find the probability that a player selects two red counters.
(This path has been drawn on the tree diagram with arrows.)
Answer:
This is and rule
as we need red
and red.
Question 2: Find the probability the two counters are different colours
This means we need:
red and blue OR blue and red
P(Red and Blue) =
P(Blue or Red) =
P(Red and Blue OR Blue and Red) =
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Exam Questions S8
1.
2.
15
3.
4.
16
5.
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Probability from Venn Diagrams
80 pupils in a certain school may choose one, two or three optional
subjects
History (H), Geography (G) and French (F).
The numbers in the Venn diagram represent the number of pupils in
each subset.
H
16
5
B
20
1
3
7
8
21
7
G
If a pupil is chosen at random from the group, find the probability that
(c) he studies Geography,
(d) he studies one optional subject only.
If it is known a pupil studies History, find the probability that
(g) he studies biology as well.
(h) he studies geography but not biology.
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