1. MODULE 15 - PROBABILITY

2. WHAT YOU WILL BE ABLE TO DO IN THIS EXCITING MODULE:
Use probability models for comparing the relative frequency of an outcome (understanding, for example, that it takes a very large number of trials before the relative frequency of throwing a head approaches the probability of throwing a head).
(LO 4 AS 2a)

3. Use Venn diagrams as an aid to solving probability problems, appreciating and correctly identifying:
the sample space of a random experiment
an event of the random experiment as a subset of the sample space the union and intersection of two or more subsets of the sample space;
P(S) = 1 where S is the sample space.
P(A or B) = P(A) + P(B) - P(A and B) where A and B are events within a sample space.
disjoint (mutually exclusive) events, and is therefore able to calculate the probability of either of the events occurring by applying the addition rule for disjoint events:
P(A or B)=P(A)+P(B)
complementary events, and is therefore able to calculate the probability of an event not occurring:
P( not A)=l - P(A)
(LO 4 AS 2b)

4. AT THE END OF THIS MODULE, DO THE FOLLOWING SELF-ASSESSMENT TO SEE WHAT YOU KNOW AND WHAT YOU ARE STILL STRUGGLING WITH (TICK OFF THE APPROPRIATE COLUMN).

5. CRITERIA
I do not understand this topic at all and need serious help.
I am sort of OK but still need to work on this topic a lot more.
I really do understand this topic completely.
Events
The probability scale
Relative frequency
Venn diagrams
Union and intersection
Mutually exclusive events
Inclusive events
Exhaustive events
Complementary events

6. The likelihood or chance of an event happening is called the probability of that event taking place. An experiment involves activities such as tossing coins or the throwing of dice. If the experiment you are doing involves the tossing of a coin, then each toss of the coin is called a trial. An outcome is the result of a trial.
Suppose that the experiment involves tossing a coin twice. You can have many trials of tossing the coin twice, however, there can only be four possible outcomes:
HEADS, HEADS
HEADS, TAILS
TAILS, HEADS
TAILS, TAILS
The set of all possible outcomes is called the sample space. We write this as follows:

7. S={HH,HT,TH,TT}
The number of outcomes in S is written as n (S) = 4.
An event is a collection of one or more outcomes of an experiment. It consists of one or more elements of the sample space.
Some possible events for the experiment above could be:
Event A = {getting a head on the first toss}
= {HH, HT}
HH and HT are called favorable outcomes.
We say that n (A) = 2 , i.e. the number of outcomes in event A is 2.
Event B = {getting three tails in any order}
= {HT, TH, TT}
We say that n (B) = 3 , i.e. the number of outcomes in event A is 3.

8. Different types of events
(a) Certain events
These events will always happen. If you toss a coin twice and the event is
E = {getting a head or a tail}, then you are certain to get a head or a tail on each toss. Each outcome in the sample space S ={HH, HT, TH, TT} will contain a head or tail. Your chances of event E happening would be 4 out of 4,
i.e. =1.

9. (b) Even chance events
Consider the experiment with S = {HH, HT, TH, TT} and the events A = {getting two heads or two tails) = {HH, TT}
A = {not getting two heads or two tails} = {HT, TH}
Event A has a 2 out of 4 chance of happening. Event B also has a 2 out of 4 chance of happening. Both events have an equal chance of happening.
We write this probability as or

10. (c) Equally likely events
Suppose that a die is thrown. The outcomes will be {1,2,3,4,5,6}.
Now consider the following events:
A = {getting the number 1} The chance is 1 out of 6 or
B = {getting the number 2} The chance is 1 out of 6 or
C = {getting the number 3} The chance is 1 out of 6 or
D = {getting the number 4} The chance is 1 out of 6 or
E = {getting the number 5} The chance is 1 out of 6 or
F = {getting the number 6} The chance is 1 out of 6 or
Each event has an equally likely chance of happening. These events are called equally likely events. The outcomes in the sample space are equally likely to happen. We say that these events are unbiased.

11. (d) Impossible events
Consider the following event:
G = {getting the number 7 from a die that is thrown} . This is impossible because the die doesn't contain the number 7. The chance of this happening is = 0 .
We say that G= or G = { }, which is called the empty set.

12. The probability scale
We write probabilities as fractions, decimals or percentages. The less likely an event is to happen, the smaller the fraction. The more likely the probability, the greater the fraction.

13. Calculating probabilities
When all outcomes of an activity are equally likely, you can calculate the probability of an event happening by using the following definition:
P(E)=
number of favorable outcomes = n (E)
_____________________________________
total number of possible outcomes = n (S)

14. (e) Elementary and composite events
Suppose that a die is thrown. The outcomes will be {1,2,3,4,5,6}.
Consider the following events:
A = {getting the number 2}
B = {getting the prime numbers} = {2,3,5}
Event A can only happen in one way. It is called an elementary event. Event B can happen in more than one way. It is called a composite event.

15. Example
Suppose that you throw a dice. The possible outcomes are S ={1,2,3,4,5,6}. If the event is E = {getting an even number} = {2,4,6}, then .
P(E)=
If the event is F = {getting a number greater than 6} = { } It is impossible to get a number greater than 6 when rolling a die.
P(F)=

16. Ways of determining probability
Theoretical probability
This method involves logical thought. If a die is thrown and the event is getting the number 4, we theoretically know that the probability of this event happening is

17. Experimental probability
This method involves many trials. Suppose you threw a die 200 times and then counted the number of times you got the number 4. You might land up getting this number 33 times. You would then say that the relative frequency of getting the
number 4 is = = 0,2 = . Only after many trials will the relative frequency get closer to the theoretical probability of .

18. Examples on calculating probabilities
Suppose that you are required to draw cards numbered from 1 to 12 out of a hat. The sample space is S = {1,2,3,4,5,6,7,8,9,10,11,12} Consider the following events:

19. A = {drawing prime numbers} = {2,3,5,7,11}
B= {drawing odd numbers} ={1,3,5,7,9,11}
C = {drawing factors of 6} = {1,2,3,6}
D = {drawing numbers greater than 12} = { }
E= {drawing a natural number} == {1,2,3,4,5,6,7,8,9,10,11,12}
Calculate the probability of each event happening:
P(A)=

20. Example 2 Consider the word MATHEMATICS
A letter is chosen from this word.
(a) Find the probability that the letter chosen is M.
There are a total of 11 letters in the word. M appears twice. Therefore the probability of getting M is
(b) Find the probability that the letter chosen is not M. The probability is .
(c) Find the probability that the letter chosen is a vowel {A,E,I,0,U}. The probability is
(d) Find the probability that the letter chosen is H. The probability is

21. EXERCISE 1
tickets were sold for a raffle. Jason bought 12 tickets. What is the probability that he:
(a) will win the prize?
(b) will not win the prize?

22. 2. There are 60 boys and 40 girls. auditioning for POP IDOLS. The
There are 60 boys and 40 girls auditioning for POP IDOLS. The names of the contestants are put into a hat and a name is drawn. What is the probability that:
(a) a boy will be selected on the first draw?
(b) a girl will be selected on the first draw?

23. A bag contains 6 blue marbles, 5 red marbles, 8 green marbles and 9 white marbles. What is the probability of:
(a) drawing a white marble?
(b) drawing a green marble?
(c) drawing a blue marble?
(d) drawing a red marble?
(e) drawing a red or blue marble?
(f) drawing a blue or green marble?
(g) drawing a pink marble?
(h) drawing a white, green or red marble?
(i) not drawing a blue marble?

24. A robot shows green for 2 minutes, amber for 30 seconds and red for 1 minute. Calculate the probability that the next motorist arriving at the intersection finds the lights:
(a) on red.
(b) on green.
(c) on amber.
(d) not on red.
(e) not on green.
(f) not on amber.

25. 5. A letter is drawn from the word. PROBABILITY
5. A letter is drawn from the word PROBABILITY. Find the probability of:
(a) drawing the letter P.
(b) drawing the letter I.
(c) drawing the letter A.
(d) drawing a vowel.
(e) drawing the letter B.
(f) not drawing a vowel.

26. A card is drawn from a pack of 52 cards
A card is drawn from a pack of 52 cards. Determine the probability of drawing:
(a) a heart.
(b) a jack of clubs
(c) an ace.
(d) a king or queen
(e) neither a heart or a spade.
A six-sided die is thrown. Determine the probability of:
(a) throwing a 6.
(b) throwing a 2 or a 4.
(c) throwing an odd number.
(d) not throwing a 5.

27. Venn Diagrams
Venn diagrams represent a sample space and its events.

28. Example 1
Consider the hat experiment where S = {1,2,3,4,5,6,7,8,9,10,11,12}.
Suppose that there are two events:
A = {drawing numbers less than or equal to 6} = {1,2,3,4,5,6}
B = {drawing numbers greater than 6} = {7,8,9,10,11,12} We can represent the sample space and its events in the following Venn Diagram:

29. There are no elements in common for event A and B
There are no elements in common for event A and B. Event A and B exclude each other. If A happens, then B cannot happen. Both cannot happen at the same time. If B happens, A cannot happen. Both cannot happen at the same trial. The outcomes for A and B are said to be disjoint. The sets don't intersect (no intersection). Events with no elements in common are called mutually exclusive events.

30. Example 2
Consider the hat experiment where S = {l, 2,3,4,5,6,7,8,9,10,11,12}. Suppose that there are two events:
C = {drawing a factor of 6} = {1,2,3,6} D = {drawing a factor of 9} = {l, 3,9} We can represent the sample space and its events in the following Venn Diagram:

31. Here we have elements in common from both sets, i. e. {1,3}
Here we have elements in common from both sets, i.e. {1,3}. The sets have an intersection. The numbers 4,5,7,8,10,11,12 are not elements of C or D., although they are still elements of the sample space, S.
Events with elements in common are called inclusive events.

32. The union and intersection of sets
Example 1
Consider the following Venn diagram:

33. The union of A and B is an event consisting of all outcomes that are in A or B.
We write the union in the following ways:
• A B = {1,2,3,4,5,6,7,8,9,10,11,12}
• A or B= {1,2,3,4,5,6,7,8,9,10,11,12}

34. Example 2
Consider the following Venn diagram:

35. The union of C and D is an event consisting of all outcomes that are in C or D. We write the union in the following ways:
• C D ={l,2,3,6,9}
• C or D ={1,2,3,6,9}
Here the numbers 4,5,7,8,10,11,12 are excluded from the union of C and D. The number 1 and 3 appear in both set C and D and are written only once in the union set.

36. Intersection Example 1 Consider the following Venn diagram:
We write the intersection in the following ways:
C D={l,3}
C and D ={1,3}

37. Example 2 Consider the following Venn diagram:
The events A and B have no outcomes in common, i.e. no intersection. We write this as follows:
A B={}or (j)
A and B = { } or (1)

38. Mutually exclusive events
In a particular experiment, events which have no elements in common are called mutually exclusive events. These events cannot happen at the same time or simultaneously.
Consider the hat experiment where S = {l,2,3,4,5,6,7,8,9,10,11,12}. Suppose that there are two events:
A = {drawing numbers less than or equal to 6} = {1,2,3,4,5,6}
B = {drawing numbers greater than 6} ={7,8,9,10,11,12} We can represent the sample space and its events in the following Venn Diagram:

39. If event A occurs, it excludes the possibility of event B occurring
If event A occurs, it excludes the possibility of event B occurring. Events A and B are based on one particular experiment, drawing a number from a hat. Clearly, we can see that
A B = {1,2,3,4,5,6,7,8,9,10,11,12}
and A B = { }

40. Therefore, it is clear that n (A B) = 12 and n (A B) = 0.
If we now consider probabilities:
Notice that:
P(A)+P(B)

41. Therefore, it is true that with mutually exclusive events:
P(A B)=P(A)+P(B) .
We can write this statement as P(A or B) = P(A) + P(B).
Also, it is clear that with mutually exclusive events, P(A B) = 0.
Alternatively, we can write this statement as P(A and B) = 0 .

42. Events that are not mutually exclusive (inclusive events)
In a particular experiment, events which do have elements in common are called inclusive events. These events can happen at the same time or simultaneously. Consider the hat experiment where
S = {l, 2,3,4,5,6,7,8,9,10,11,12} and the events happening are:
C = {drawing a factor of 6}
= {1,2,3,6}
D = {drawing a factor of 9}
= {1,3,9} We can represent the sample space and its events in the following Venn Diagram:

43. Here, both events can happen at the same time
Here, both events can happen at the same time. If the number 1 or 3 is drawn from
the hat, then both event C and D happened at the same time.
C D={l,2,3,6,9} and C D={l,3}.
Therefore, n (C D) = 5 and n (C D) =2
Also n (C) = 4 and n (D) = 3
Clearly, n (C) + n (D) n (C D)
However, n (C) + n (D) – n (C D) =
= = 5 = n (C D)
Therefore, it is true that:
n (C D)= n (C) + n (D) – n (C D).

44. Alternatively, we can write this statement as follows:
If we now consider probabilities:
Notice that:
P(C) + P(D) - P(C D)
Therefore, it is clear that:
P (C D) P(C)+P(D).
However, it is clear to see that:
P(C D) = P(C) + P(D) - P(C D)

45. Summary For two mutually exclusive events A and B, the following rules are true:
P(A B) = P(A) + P(B)
P (A B)= 0 Alternative notation:
P (A or B)=P(A)+P(B)
P(A and B) = 0
For non-mutually exclusive (inclusive) events, the following rules are true:
P (A B) P(A) + P(B)
P(A B) = P(A) + P(B) - P(A n B)
Alternative notation:
P(A or B) P(A) + P(B)
P(A or B) = P(A) + P(B) - P(A and B)

46. Exhaustive events
Two events are said to be exhaustive if, together, they cover all elements of the sample space. For example, suppose that the sample space is S ={1,2,3,4,5,6} and the following are events:
A ={1,3,5}, B ={2,4,6}, C={2}, D={3,6}
A and B are exhaustive events because together they cover all elements of the sample space S.
Also:
A B= {1,2,3,4,5,6} and A B={ }.

47. P(A)=
P(B)=
P(A)+P(B)=
And P(A B)=
P(A B)=P(A)+P(B)
Also P(A B) = 0
Therefore A and B are mutually exclusive.
So A and B are mutually exclusive, exhaustive events.
A and C are not exhaustive since together they do not cover all elements of S.
Also:
A C={1,2,3,5} S and A C={}
P(C)=
P(A)+P(C)=
and P(A C) =
P(A C)=P(A)+P(C)
P(A C)=0
Therefore A and C are mutually exclusive.

48. So A and C are mutually exclusive, but not exhaustive events.
A and D are not exhaustive since together they do not cover all elements of S. Also:
A D={l,3,5,6} S
and
A C={3}
P(A)=
P(D)=
P(A)+P(D)=
P(A D)=
P(A D) P(A) + P(D)
Also P(A D)=

49. Now:
P(A) + P(D) - P(A D) =
= P(A D)
P(A D) = P(A) + P(D) - P(A D)
Therefore A and D are inclusive.
So A and D are inclusive, but not exhaustive events.

50. Complementary events
Mutually exclusive, exhaustive events are called complementary events.
Example
A bag contains 5 red smarties, 2 green smarties, 3 pink smarties and 4 blue smarties.
Suppose that event A = {getting a blue smartie}.
The event {not getting a blue smartie} is called the complement of event A.
We may refer to the complement of event A as
{not A}. The complement set is written as
A' = {not getting a blue smartie} .

51. A and A' are mutually exclusive because there are no elements in common. We can represent the information in a Venn Diagram:

52. Notice the following:
A A'
= {B 1 ,B2,B3,B4,G1 ,G2,P1 ,P2,P3,R 1 ,R2,R3,R4,R5}
A A'={ }
P(A)=
P(A')=
P(A A') =
Therefore P(A A') = P(A) + P(A')
And P(A A')=
Therefore, A and A' are mutually exclusive and exhaustive. In other words, they are complementary events.
From the above it can be deduced that:
P(A) + P(A')= 1 or P(A') = 1 - P(A)
Therefore for any two complementary events:
P(A') = 1 - P(A)
Alternative notation:
P (not A) = 1 - P(A)

53. EXERCISE 2
Cards numbered from 1 to 12 are put into a box and shaken. Cards are then drawn and replaced.
The following events are given:
A = {drawing an even number}
B = {drawing an odd number}
C = {drawing a number greater than 7}
D = {drawing a number less than 5}
E = {drawing natural numbers less than 7}
F = {drawing natural numbers greater than 4}

54. (a) Draw a Venn Diagram to represent events A and B.
(b) Determine P(A B)
(c) Determine P(A B)
(d) Show by using the rules that A and B are mutually exclusive and exhaustive.
(e) Are events A and B complementary? Verify your answer by using the rule for complementary events.
(f) Draw a Venn Diagram to represent events A and C.
(g) Determine P(A or C)
(h) Determine P(A and C)
(i) Show by using the rules that A and C are inclusive but not exhaustive.
(j) Are events A and C complementary? Verify your answer by using the rule for complementary events.
(k) Draw a Venn Diagram to represent events C and D.
(1) Determine by using the rules, whether C and D are mutually exclusive or inclusive, exhaustive or not exhaustive, complementary or not complementary.
(m) Draw a Venn Diagram to represent events E and F.
(n) Determine by using the rules, whether E and F are mutually exclusive or inclusive, exhaustive or not exhaustive, complementary or not complementary.

55. 2. G and H are inclusive events in a sample space S.
If it is given that P(G or H) = P(G) = ,P(G) = and P(H) = determine:
(a) P(G H) if n (S) = 20.
(b) P (not G)

56. 3. A tennis player won her matches over three sets.
She had the following number of aces per set. A set consists of at least 6 games.
Use these results to find the probability that the winner hit an ace:
(a) in the first set.
(b) in the second set.
(c) in the third set.
(d) in the match.
Set 1 2 3
Aces 10 8 11