Statistics For Social Sciences – MATH1208
Unit 5 – Probability
Probability is a measure of how likely
something will occur.
The action of collecting data with outcomes which are recordable is called an experiment. An experiment could be ‘the count of the
number of students who miss statistics class each week or a record of the colour of the cars parked at Portmore Community College.
Each throw of a die is known as an experiment
or trial which can be repeated. The result (one
An event is a collection of one or more outcomes of an experiment.
The sample space, S, is the set of all possible
outcomes of an experiment. The sample space is normally represented using brackets.
Consider rolling a single, six sided die and recording the number of dots on the top side. This experiment has six possible outcomes, so the sample space is S = {1, 2, 3, 4, 5, 6}.
Exercise
Write down the sample space for the following experiments.
2. The tossing of two fair coins once. 3. The tossing of two fair dice once.
4. The spinning of two different spinners once, as pictured below.
5. The tossing of a red and a blue dice once.
The Probability of an Event
The probability of an event ‘A’ occurring, is the number of possible ways in which the event ‘A’ can occur divided by the total number of possible outcomes in the sample space, where
1 2 3
each outcome is equally likely to occur. That is,
P(A) = .
The Law of Large Numbers
The law of large numbers says that as the number of replications of an experiment
increases, the estimate of the probability of an event gets closer to the true or real probability. The probability of obtaining a 4 when I throw a fair die is . On the other hand, if we are not sure that the die is fair, the probability could be arrived at by conducting several throws and
estimated probability or empirical probability of an event is taken as the relative frequency of the event when the number of observation is large.
The probability of an event ranges from 0 (the impossible event) to 1 (the certain event). That is, 0 P(A) 1.
The complement of an event A or not A,
denoted A1 or , is the set of all outcomes in
the sample space, S, that do not correspond to the event A. U
A1
Since the event A is the set of outcomes in S and the complement of the event, A1, is the set
of all outcomes in S that do not correspond to A, then P(A) + P(A1) = 1. Thus, the total
probability of an event is 1. Furthermore, P(A) = 1 – P(A1) and P(A1) = 1 – P(A).
The complement of A, A1, is all the numbers
not greater than 2. Thus, P(A1) = , since two
outcomes corresponds to A1.
Hence, P(A) = 1 – P(A1), that is,
P(A) = 1 – = .
The general addition rule is
P(A B) = P(A) + P(B) – P(A B) P(A B) = P(A) + P(B) – P(A B).
Laws of Probability – OR and AND
the two events. Union is associated with ‘addition’.
For example, if a fair die is rolled, where event A is the number that comes up is even and
event B is a 3, then the probability that the number on top is even or a 3 is: P(A B) = + = .
The event ‘A and B’ (A B) is the event that A and B both occurs. It is often referred to as the intersection of the two events. Intersection is associated with ‘multiplication’.
For example, a fair coin is tossed and an
The sample space can be represented in a table as follows:
1 2 3 4 5 6
H H1 H2 H3 H4 H5 H6
T T1 T2 T3 T4 T5 T6
The total number of possible outcomes in the sample space S is n(S) = 12.
The probability that a head on the coin and a six on the die occurs is .
The probability that a tail on the coin and an odd number on the die occurs is .
1. A bag contains 60 marbles of which 15 are red and the remainders of marbles are green. a. How many green marbles are in the bag? b. What is the probability of drawing a green marble?
c. What is the chance of drawing a red marble? d. What is the chance of drawing a yellow
marble?
e. What is the probability of drawing either a red marble or a green marble?
2. A teacher chooses a student at random from a class of 30 girls. What is the probability that the student chosen is a girl?
3. A glass jar contains 5 red, 3 blue and 2 green jelly beans. If a jelly bean is chosen at random from the jar, what is the probability that it is not blue?
4. An ordinary, fair, 6-sided die is rolled. What is the probability of rolling:
a) 4? b) a number less than 3? c) an even number?
e) prime number? f) a composite number? g) a multiple of 3?
5. The number of goals scored by a team in 30 matches was recorded in the frequency table below.
What is the probability that the number of goals scored in the 30 matches is:
a. 4 b. less than 3 c. 4 or more
6. A card is chosen from a standard pack of 52 playing cards. What is the probability that the card is: Take cards to class
Number of Goals
0 1 2 3 4 5
a) an Ace? b) a red card? c) a black King? d) a clubs?
7. In a class of 25 students, 17 study English, 15 study Physics and 11 study both of these subjects. Find the probabilty that a student chosen at random:
a. studies Physics.
b. studies English only.
c. studies both Physics and English.
d. does not study either English or Physics. e. study Physics or English
9. A paper bag contains eight (8) green
marbles and three (3) red marbles and a plastic bag contains four (4) green marbles and six (6) red marbles. If a marble is drawn from each bag, what is the probability that both are red?
Ans:
10. Find P(X Y), given that P(X) = ,
P(Y) = and P(X Y) = . Ans:
Mutually Exclusive Events
That is, the intersection of A and B is empty (A B = ). Thus, for mutually exclusive events the P(A B) = P(A) + P(B) and
P(A B) = 0.
For example, the event that James walks to
school and the event that he cycles to school at the same time are mutual exclusive events.
Also the event that the ADSW Year 1 students attends Statistics and FCM 2 classes during the period 1 to 5 pm on Tuesdays.
Another example is the event of turning left or right.
Two events A and B are non-mutually
exclusive, if they can occur simultaneously (together). That is, the intersection of A and B is not empty. Thus, for non-mutually exclusive events the P(A B) = P(A) + P(B) – P(A B). For example, the events of tossing a coin and the rolling of a die are non-mutually exclusive events.
Also the events of rolling a pair of dice and obtaining a total of 5 or a prime number.
Independent Events
non-occurrence of the other event. For example, the events of tossing a coin or rolling a die
together. Thus for an independent event P(A B) = P(A) P(B).
Dependent Events
Two events A and B are dependent events if the occurrence or non-occurrence of one event
affects the occurrence or non-occurrence of the other event. For example, the probability of
drawing a marble from a bag containing marbles, without replacement.
Note: De Morgan’s law (A B)1 = A1 B1
(A B)1 = A1 B1 (A B)1 = A1 B1
Exercise
Answer the following.
Two events A and B are such that P(A1) =
and P(B) = . If A and B are independent events, find:
a) P(A) Ans:
c) P(A B1) Ans:
d) P(A1 B1) Ans:
Exercise
Answer the following.
1. Given that A and B are two events.
Determine, in each case below, if events A and B are mutually exclusive events.
a. P(A) = , P(B) = and P(A B) = .
2. Given that G and M are two events. If P(G) = and P(M) = , find :
a. P(M G) and P(G M) if M and G are mutually exclusive events.
Ans: P(M G) = 0 , P(G M) =
b. P(M G) and P(G M) if M and G are
independent events.
Ans: P(G M) = and P(M G) =
a. either walks or cycles to school.
Ans: P(W C) =
b. neither walks nor cycles to school.
Ans: P(W C)1 =
c. walks and cycles to school.
Ans: P(W C) = 0
4. Events A and C are independent. Given that P(A) = , P(B) = , P(A C) = and
P(C B) = .
5. Given that A and B are two events such that
P(A) = , P(B) = and P(A B) = , find:
a. P(A B) Ans: P(A B) =
b. P(A1 B1) Ans: P(A1 B1) =
c. P(A1 B1) Ans: P(A1 B1) =
6. The table below shows the distribution of students in two areas of study relative to their gender.
GENDER Calculus Statistics TOTAL
Male 170 160 330
TOTAL 400 300 700
a. What is the probability that a student selected at random is a male? Ans:P(male) =
b. What is the probability that a student
selected at random is a Statistics student given that the student is a female?
Ans:P(statistics student/female) =
c. What is the probability that a student selected at random is a Calculus student?
d. What is the probability that a student
selected at random is a Calculus student or a male student?
Ans:P(calculus or male student) =
7. A bag contains 10 marbles numbered 1 to 10 inclusive. A ball is selected at random. What is the probability that the number is:
a. prime? Ans: P(prime number) =
b. even or greater than 4?
Ans: P(even or > 4) =
a. Find the probability that neither A nor B occurs. Ans: P(A B)1 = 0.3
b. Find the value of P(B) for which A and B are mutually exclusive. Ans: P(B) = 0.3
c. Find the value of P(B) for which A and B are independent, if P(A and B) = 0.2. Ans: P(B) = 0.5
9. A bag contains three green marbles and two red marbles. In an experiment two marbles are drawn from the bag. Identify the sample space if the first marble is replaced.
R2 R2 R1 R2 R2 R2 G1 R2 G2 R2 G3
G1 G1 R1 G1 R2 G1 G1 G1 G2 G1 G3
G2 G2 R1 G2 R2 G2 G1 G2 G2 G2 G3
G3 G3 R1 G3 R2 G3 G1 G3 G2 G3 G3
10. At a car repair shop 50% of the cars have an engine problem, 43% of the cars have a
transmission problem and 17% of the cars both have an engine and transmission problem. Find the probability that a car chosen at random has: a. an engine or transmission problem.
Ans: P(E T) = 0.76
b. no transmission or engine problem.
Ans: P(E T)1 = 0.24
Conditional probability is the probability that an event will occur given that another event has already occurred. For two events A and B, the conditional probability that A occurs given that B has already occurred is written as P(A/B).
Thus, P(A/B) = .
Also P(A B) = P(A/B) . P(B), P(A B) = P(B/A) . P(A)
and P(A1/B) = 1 – P(A/B).
P(B) = P(B A) + P(B A1) and
P(A) = P(A B) + P(A B1)
Furthermore, if A and B are mutually exclusive
events then
P(B) = P(B/A) . P(A) + P(B/A1) . P(A1),
P(B) = P(B A1).
P(A) = P(A/B) . P(B) + P(A/B1) . P(B1),
P(A) = P(A B1).
Exercise
1. Three fair coins are tossed. What is the probability of recording at least two heads,
given that the first coin was recorded as heads?
Ans: P(at least 2H/first is H) =
2. A bag contains 7 black markers and 5 red markers. Find the probability of randomly drawing two black markers consecutively, if the marker is:
a. replaced after each draw.
b. not replaced after each draw.
Ans: P(black and black) =
3. a) Draw a Venn diagram to show the n(A) = 15, n(B) = 12, n(A B) = 7 and n(A B)1 = 15.
Calculate:
i. P(A/B) Ans: P(A/B) =
b) Draw a Venn diagram to show the n(D) = 12, n(F) = 15, n(D F) = 0 and n(D F)1 = 8.
Calculate:
i. P(D/F) Ans: P(D/F) = = 0 ii. P(F/D) Ans: P(F/D) = = 0
4. The events A and B are such that P(A) = ,
P(B) = and P(A B) = . Find :
a. P(A B) Ans: P(A B) = or
5. Two events A and B are known to be
mutually exclusive. P(A) = 0.4 and P(B) = 0.25. Find:
a. P(A and B) Ans: P(A B) = 0 b. P(A or B) Ans: P(A B) = 0.65
6. Suppose that A and B are events and we know that P(A) = 0.7, P(B) = 0.5 and P(A B) = 0.3.
Calculate:
a. P(A1) Ans: P(A1) = 0.3
7. C and D are events such that P(C D) = 0.8, P(C D) = 0.3 and P(C/D) = 0.7. Find:
a. P(D) Ans: P(D) =
b. P(D1) Ans: P(D1) =
c. P(C) Ans: P(C) =
d. P(D1/C) Ans: P(D1/C) = e.
P(C1/D) Ans: P(C1/D) = 0.3 f.
A tree diagram (probability tree diagram) is a chart that can be used to list all possible
outcomes of a conditional probability.
Initially, two more branches (paths) spring from a root (single point).
The outcome is stated at the end of each branch.
The probability of an outcome is stated on each branch
The probability for an outcome is obtained by multiplying the probabilities along the branch.
The sum of the probabilities for all
obtained by adding the outcomes at the end of each path.
NOTE: The sum of the probabilities of an event A is one. That is, P(A) + P(A1) = 1.
Exercise
Answer the following.
1. A fair coin is tossed two times and the outcomes recorded.
a. Draw a tree diagram to represent all possible outcomes.
i. P(H and H) Ans: P(H and H) =
ii. P(H and T) Ans: P(H and T) =
iii. P(T and H) Ans: P(T and H) =
iv. P(T and T) Ans: P(T and T) =
c. Calculate the sum of the probabilities for all outcomes. Ans: 1
[Unit 2 Text - Toolsie, R., Page 571]
he takes route B, then the probability of being late is 10%.
a. Draw a tree diagram to represent all possible outcomes.
b. Find the probability that Robert is late for school.
Ans: P(L) = P(A and L) or P(B and L) = P(A
L) P(B L) = 0.06 + 0.02 = 0.08
c. Given that Robert is late for school, find the probability that he took route B.
Ans: P(B/L) = 0.75
e. Given that Robert is not late for school, find the probability that he took route A.
Ans: P(A/L1) = 0.41
[Unit 2 Text - Toolsie, R., Page 573] 3. A container has 7 cartridges of black printer ink and 5 cartridges of red printer ink. Two
cartridges are picked at random, one after the other, from the container.
a. Draw a tree diagram to illustrate all possible outcomes if the cartridge is:
i. replaced ii not replaced
i. black followed by red
Ans: Replaced P(B R) =
Not replaced P(B R) =
ii. of different colours
Ans: Replaced P(B R) + P(R B) =
Not replaced P(B R) + P(R B) =
iii. Both the same colour.
Replaced P(B B) + P(R R) =
Not replaced P(B B) + P(R R) =
4. A teacher notices that in class, 65% of the students complete their homework regularly. Of those who complete their homework 80% do well in tests, while only 45% of those who do not complete their homework do well in tests.
a. Represent this information on a tree diagram b. Hence, find the probability that a student chosen at random did well on the most recent test. Ans: 0.68
5. Ray either walks or travels by bus to school. If it rains, the probability that he takes the bus is , whereas if it’s fine the probability that he takes the bus is . The probability of rain on any day is .
a. Draw a probability tree diagram to represent the information above.
b. What is the probability that on a day chosen at random he takes the bus. Ans:
6. A bag contains 5 red marbles and 3 blue
marbles. Two marbles are randomly withdrawn from the bag consecutively.
a. Represent the tree diagram, for the
information above if the marble is replaced.
i. Determine the probability of drawing a red marble. Ans:
ii. What is the probability of drawing the same colour? Ans:
b. Represent the tree diagram, for the
i. Determine the probability of drawing a red marble. Ans:
ii. What is the probability of drawing the same colour? Ans:
[Unit 2 Text - Campbell, E., Page 191]
7. A truck is used for two trips each day. On each trip the truck carries a heavy load (H) with probability 0.25. Assuming that the load carried in the second trip of the day is independent of the previous load:
b. What is the probability that the truck carries just one load on a particular day?
[Text - Francis, A., Page 430]
8. A firm is independently working on two
separate jobs. There is a probability of only 0.3 that either of the jobs will be finished on time. a. Draw a probability tree diagram for this
information.
b. Find the probability that:
i. both ii. neither iii. just one iv. at least one
9. Box A contains 7 red cricket balls and 3 white cricket balls. Box B contains 4 red
cricket balls and 5 white cricket balls. A cricket ball is drawn at random from Box A and placed in box B. Box B is thoroughly shaken and then a cricket ball is drawn at random from it.
a. Draw a probability tree diagram illustrating all possible outcomes of this experiment.
b. Use the probability tree diagram to find the probability that:
ii. a white cricket ball is drawn from box B. [Unit 2 Text - Toolsie, R., Page 576]
10. Two vacuum cleaner salesmen ‘A’ and ‘B’ must each make two calls per day, one in the morning and one in the afternoon. ‘A’ has
probability 0.4 of selling a cleaner on any call, while ‘B’ has a probability 0.1 of a sale. ‘A’ works independently of ‘B’ and for each
salesman, morning and afternoon results are independent of each other. Find the probability that, in one day:
a. ‘A’ sells two cleaners
c. ‘B’ makes at least one sale
