1. Probability
Joan Ridgway

2. Getting a C for GCSE Maths
Throwing a 6 on a dice
Tossing a coin and …getting “heads”
It will rain tomorrow
Winning the lottery
Impossible
Certain
All probabilities lie somewhere on a scale between “Impossible” and “Certain”

3. Getting a C for GCSE Maths
Throwing a 6 on a dice
Tossing a coin and …getting “heads”
It will rain tomorrow
Winning the lottery
Impossible
Certain 1
All probabilities lie somewhere on a scale between “Impossible” and “Certain”
The probability scale goes from 0 to 1

4. 1 0.5 0.7 Getting a C for GCSE Maths Throwing a 6 on a dice
Tossing a coin and …getting “heads”
It will rain tomorrow
Winning the lottery
Impossible
Certain
1 14,000,000 16 1
0.5
0.7
95%
Probabilities can be expressed either as fractions or as decimals (and sometimes as percentages)

5. The probability of throwing a 6 with a fair dice is16
P(6) = 16
So the probability of not throwing a 6 is 56
P(not 6) = 1- 16 = 56

6. P(rain) = 0.7 P(not rain) = 1 – 0.7 = 0.3
If the probability that it will rain tomorrow is 0.7 …..
P(rain) = 0.7
Then the probability that it will not rain tomorrow is 0.3
P(not rain) = 1 – 0.7 = 0.3

7. 0.5, ½ or 50% Suppose I toss a coin:
What is the probability of getting a head?
0.5, ½ or 50%

8. or Suppose I toss two coins:
If I toss the coin twice, I would get one of these combinations:
Heads, Heads Heads, Tails Tails, Heads Tails, Tails
H, H H, T T, H T, T or
What is the probability of getting two heads?
Only one of these four combinations is two heads

9. or Suppose I toss two coins:
If I toss the coin twice, I would get one of these combinations:
Heads, Heads Heads, Tails Tails, Heads Tails,Tails
H H
H, H H, T T, H T, T or
What is the probability of getting two heads?
Only one of these four combinations is two heads
So the probability of getting a two heads in a row is ¼

10. T, T T, H T H, T H, H H Second Coin First Coin
A Sample Space is a list of all the possible outcomes, e.g. HH, HT, TH, TT
We can show this in a Sample Space Diagram:
T, T
T, H T
H, T
H, H H
Second Coin
First Coin
There are 4 possible outcomes if you toss a coin twice
So the probability of two heads is ¼

11. Suppose I throw a die.
There are 6 equally likely outcomes.

12. Suppose I throw two dice.

13. Suppose I throw two dice.
We can show the possible outcomes in a Sample Space Diagram:
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice
There are 36 (6 x 6) possible outcomes if you throw two dice.

14. If you throw two dice, what is the probability of getting a “double”?
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice

15. If you throw two dice, what is the probability of getting a “double”?
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice
6 out of the 36 possible outcomes are “doubles”, so the probability is
6 36

16. If you throw two dice, what is the probability of getting a “double”?
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice
6 out of the 36 possible outcomes are “doubles”, so the probability is
1 6

17. What is the probability of scoring 9 or more?
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice

18. What is the probability of scoring 9 or more?
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice
10 out of the 36 possible outcomes add up to 9 or more, so the probability is

19. What is the probability of scoring 9 or more?
6, 6
6, 5
6, 4
6, 3
6, 2
6, 1 6
5, 6
5, 5
5, 4
5, 3
5, 2
5, 1 5
4, 6
4, 5
4, 4
4, 3
4, 2
4, 1 4
3, 6
3, 5
3, 4
3, 3
3, 2
3, 1 3
2, 6
2, 5
2, 4
2, 3
2, 2
2, 1 2
1, 6
1, 5
1, 4
1, 3
1, 2
1, 1 1
Second Dice
First Dice
10 out of the 36 possible outcomes add up to 9 or more, so the probability is
5 18

20. We cannot always calculate the probability of an event; sometimes we have to estimate it.
Suppose we did not know whether a dice was fair or weighted. We could throw it 100 times to find how often we threw a 6.
We would expect to get a 6 about once in every 6 throws, as the probability should be 1 in 6.

21. llll llll llll llll llll llll llll llll llll ll llll llll lll
We cannot always calculate the probability of an event; sometimes we have to estimate it.
Suppose we did not know whether a dice was fair or weighted. We could throw it 100 times to find how often we threw a 6. 6 5 4 3 2 1 35
llll llll llll llll llll llll llll 12
llll llll ll 13
llll llll lll 15
llll llll llll
Frequency
Tally
100
We threw 35 sixes out of a total of 100 throws

22. 35 100 We threw 35 sixes out of a total of 100 throws
The relative frequency of throwing a 6 with this dodgy dice is:

23. . 35 100 1 6 The dice is probably weighted!!!
We threw 35 sixes out of a total of 100 throws
The relative frequency of throwing a 6 with this dodgy dice is: or
The probability of throwing a 6 with a normal dice is: or .
1 6
We threw a 6 more than twice as often as would be expected.
The dice is probably weighted!!!

24. Go back to the coins.
Remember there were 4 possible outcomes if I toss 2 coins
Heads, Heads Heads, Tails Tails, Heads Tails,Tails
H H H T T H T T or
There are 4 possible outcomes because 2 x 2 = 4, just as for two dice there are 36 possible outcomes because 6 x 6 = 36

25. There are 8 possible outcomes because 2 x 2 x 2 = 8
If I toss three coins, what are the possible combinations?
T T T
– 0 Heads (3 Tails)
There are 8 possible outcomes because x 2 x 2 = 8
H T T
1 Head (2 Tails)
T H T
T T H
H H T
H T H
2 Heads (1 Tail)
T H H
H H H
– 3 Heads

26. If I toss three coins, what are the possible combinations?
T T T
– 0 Heads (3 Tails)
The probability of 0 Heads is
1 8
H T T
1 Head (2 Tails)
T H T
The probability of 1 Head is
3 8
T T H
H H T
H T H
2 Heads (1 Tail)
The probability of 2 Heads is
3 8
T H H
H H H
– 3 Heads
The probability of 3 Heads is
1 8

27. If I toss three coins, what are the possible combinations?
The probability of 3 Heads is
1 8
H H H
– 3 Heads
H H T
H T H
2 Heads (1 Tail)
The probability of 2 Heads is
3 8
T H H
H T T
1 Head (2 Tails)
T H T
The probability of 1 Head is
3 8
T T H
T T T
– 0 Heads (3 Tails)
The probability of 0 Heads is
1 8
Simulation Hyperlink

28. Sometimes it is helpful to draw tree diagrams.
If I toss a coin twice, what is the probability of getting at least one head?
First coin
Second coin
Heads
Tails
1 .2
P(HH) =
Heads
Tails
1 .2
P(HT) =
P(TH) =
P(TT) =
Check that the probabilities add up to 1.

29. The Watsons regard one boy and one girl as the ideal family
The Watsons regard one boy and one girl as the ideal family. What is the chance of getting one boy and one girl in their planned family of two?
First Child
Second Child
Combined G B
P(G,G) = 0.5 x 0.5 = 0.25
0.5 G B
0.5
P(G,B) = 0.5 x 0.5 = 0.25
P(B,G) = 0.5 x 0.5 = 0.25
P(B,B) = 0.5 x 0.5 = 0.25
Total = 1.00
The probability of getting one of each is:
P(G,B) + P(B,G) = = 0.5

30. Sometimes it is helpful to draw tree diagrams.
Joan travels to work on her bicycle. She has to go through two sets of traffic lights on her way. At the first set of lights, the probability that they will be green is 7/10. At the second set the probability that they will be green is 3/5

31. Sometimes it is helpful to draw tree diagrams.
Joan travels to work on her bicycle. She has to go through two sets of traffic lights on her way. At the first set of lights, the probability that they will be green is 7/10. At the second set the probability that they will be green is 3/5
What is that probability that both sets will be green?
What is the probability that she will have to stop once and only once at a set of lights?
First set
Second set
Green
Not Green
3 .5
2 .5
P(both green) =
Green
Not Green
7 10
3 10
P(green, not green) =
P(not green, green) =
P(both not green) =
Check that the probabilities add up to 1.

32. Sometimes it is helpful to draw tree diagrams.
Joan travels to work on her bicycle. She has to go through two sets of traffic lights on her way. At the first set of lights, the probability that they will be green is 7/10. At the second set the probability that they will be green is 3/5
What is that probability that both sets will be green?
What is the probability that she will have to stop once and only once at a set of lights?
First set
Second set
Green
Not Green
P(both green) =
7 10
Green
Not Green
2 .5
P(green, not green) =
3 .5
3 10
P(not green, green) =
2 .5
P(both not green) =
You should simplify the probabilities where you can.

33. Sometimes it is helpful to draw tree diagrams.
Joan travels to work on her bicycle. She has to go through two sets of traffic lights on her way. At the first set of lights, the probability that they will be green is 7/10. At the second set the probability that they will be green is 3/5
What is that probability that both sets will be green?
What is the probability that she will have to stop once and only once at a set of lights?
First set
Second set
Green
Not Green
P(both green) =
7 10
Green
Not Green
2 .5
P(green, not green) =
3 .5
3 10
P(not green, green) =
2 .5
P(both not green) =
1) P(both green) =

34. Sometimes it is helpful to draw tree diagrams.
Joan travels to work on her bicycle. She has to go through two sets of traffic lights on her way. At the first set of lights, the probability that they will be green is 7/10. At the second set the probability that they will be green is 3/5
What is that probability that both sets will be green?
What is the probability that she will have to stop once and only once at a set of lights?
First set
Second set
Green
Not Green
P(both green) =
7 10
Green
Not Green
2 .5
P(green, not green) =
3 .5
3 10
P(not green, green) =
2 .5
P(both not green) =
2) P(only stopping once) =

35. Independent and Dependent Events
Two events are independent if they have no effect on each other.
If you choose two items with replacement the events will be independent because the second choice will not be affected by the first choice.
There are 10 crayons in a bag; 5 blue ones, 3 red ones and 2 green ones. I pick one at random, then put it back and choose another.
Draw a tree diagram to show the probabilities.

36. First Crayon Second Crayon Blue Red Blue Green Blue Red Red Green Blue
5 10
Blue
3 10
Red
Blue
2 10
Green
5 10
5 10
Blue
3 10
3 10
Red
Red
2 10
Green
2 10
5 10
Blue
3 10
Red
Green
2 10
Green

37. What is the probability that both crayons will be green?
First Crayon
Second Crayon
5 10
Blue
3 10
Red
Blue
2 10
Green
5 10
5 10
Blue
3 10
3 10
Red
Red
2 10
Green
2 10
5 10
Blue
3 10
Red
Green
2 10
Green

38. What is the probability that I will choose one blue crayon and one red crayon? (in either order)
First Crayon
Second Crayon
5 10
Blue
3 10
Red
Blue
2 10 or
Green
5 10
5 10
Blue
3 10
3 10
Red
Red
2 10
Green
2 10
5 10
Blue
3 10
Red
Green
2 10
Green

39. What is the probability that I will choose one blue crayon and one red crayon? (in either order)
First Crayon
Second Crayon
5 10
Blue
3 10
Red
Blue
2 10 or
Green
5 10
5 10
Blue
3 10
3 10
Red
Red
2 10
Green
2 10
5 10
Blue
3 10
Red
Green
2 10
Green

40. Independent and Dependent Events
Two events are dependent if one event will affect the other. The probability is said to be conditional.
If you choose two items without replacement the events will be dependent because the first item you choose cannot be chosen again.
There are 10 jellybabies in a bag; 5 red ones, 3 green ones and 2 yellow ones. I pick one at random and eat it, and then I choose another.
Draw a tree diagram to show the probabilities.

41. First Jellybaby Second Jellybaby Red Green Red Yellow Red Green Green
5 10
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

42. First Jellybaby
Second Jellybaby
Red
Green
Red
BE CAREFUL!!
Yellow
5 10
If I choose a red one first and eat it, there will be only 9 jellybabies left, and only 4 will be red ones!!
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

43. First Jellybaby Second Jellybaby Red Green Red Yellow Red Green Green
Red
Green
Red
Yellow
5 10
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

44. First Jellybaby
Second Jellybaby
Red
Green
Red
BE CAREFUL!!
Yellow
5 10
If I choose a green one first and eat it, there will be only 9 jellybabies left, and only 2 will be green ones!!
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

45. First Jellybaby Second Jellybaby Red Green Red Yellow Red Green Green
Red
Green
Red
Yellow
5 10
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

46. First Jellybaby
Second Jellybaby
Red
Green
Red
BE CAREFUL!!
Yellow
5 10
If I choose a yellow one first and eat it, there will be only 9 jellybabies left, and only 1 will be a yellow one!!
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

47. What is the probability that both jellybabies will be yellow?
First Jellybaby
Second Jellybaby
Red
Green
Red
Yellow
5 10
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

48. What is the probability that I will choose (and eat) one red jelly baby and one green jelly baby? (in either order)
First Jellybaby
Second Jellybaby
Red
Green
Red or
Yellow
5 10
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow

49. What is the probability that I will choose (and eat) one red jelly baby and one green jelly baby? (in either order)
First Jellybaby
Second Jellybaby
Red
Green
Red or
Yellow
5 10
Red
3 10
Green
Green
Yellow
2 10
Red
Green
Yellow
Yellow