[7] INTRODUCTION TO PROBABILITY
1) In general terms, probability is the branch of mathematics in charge of measuring the
likelihood of occurrence of given events.
2) Examples: (a) How likely is it to land Heads when tossing a coin once?
(b) How likely is it to get an ace when drawing one card from a wellshuffled deck of playing cards?
(c) How likely is it that a randomly a classmate of yours is from Florida?
(d) Suppose you own 200 shares of Microsoft Co. How likely is it for the
share price to increase by the end of trading tomorrow?
3) DEFINITIONS: Experiments, Trials, Outcomes, Random Experiments, Sample Spaces
EXAMPLES OF SAMPLES
3.1) Toss a coin once: S = {H, T}; twice: S = {HH, HT, TH, TT}
thrice: S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
four times: S={HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, THHT,
HTTH, THTH, TTHH, HTTT, THTT, TTHT, TTTH, TTTT}
3.2) Roll a single six-sided die once: S = {1, 2, 3, 4, 5, 6,}
3.3) Roll a single six-sided die twice: S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
3.4) Standard Deck of 52 playing cards: S = {Clubs: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K,
Spades: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K,
Diamonds: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K,
Hearts: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}
The twelve cards J, Q, K from the four suits form the group of face cards. There are 26
black cards (Clubs, Spades) and 26 red cards (Diamonds, Hearts).
3.5) a) An experiment consists of tossing a coin once and then rolling a single die once.
Write the complete sample space S for this experiment.
b) An experiment consists of selecting a number from the set {7, 8, 9} and then
choosing a letter from the name ASTRID. Write the complete sample space S.
4) Describe: EVENTS, SIMPLE EVENTS, COMPOUND EVENTS
5) a) Classical Approach to Assigning Probability Values to Events, E
5.1) If a balanced coin is tossed thrice, find the probability of getting: a) exactly one
head; b) no heads; c) at least one head; d) at most one head. The sample space is:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
5.2) A balanced die is rolled once. Find the probability of getting: a) a number less than 3;
b) at most a 3; c) at least a 3; d) more than 3; e) not a 3.
5.3) One card is drawn from a well-shuffled standard deck of 52 playing cards. Find
the probability of drawing: a) a face card; b) an ace; c) a number between 2 and 9.
b) Relative Frequency Approach to Assigning Probability Values to Events, E
6) A coin is tossed 50 times and the number of heads recorded is 40. Find the probability
of getting heads in this experiment?
7) Suppose that in a group of 200 patients who used the drug Lipitor, 120 experienced a
headache. If one patient is selected at random from this group, what is the probability
that s/he experienced a headache?
8) During the 2008-2009 U.S. financial crisis, many men who lost their jobs had great
difficulties finding a new job in their respective fields of education, knowledge, and/or
experience. Many single moms and families with boys employed more and more the
services of a “manny” (a male nanny), so in the absence of a job, many unemployed
men became mannies and since then, there is an increasing trend in the employment of
mannies. Suppose a group of 2014 families of a community employed nannies and
417 had mannies. Find the probability that a manny was not employed in this group.
9) In a group of 20 people, there are 11 Democrats, 7 Republicans, and 2 Independents.
If one person is selected at random from this group, find the probability that the
selected person is: a) an Independent; b) not a Republican.
c) Subjective Approach to Assigning Probability Values to Events, E
10) If in the past ten years, a particular set of meteorological conditions occurred on 500
days, of which 200 were rainy days, then what is your prediction of the probability of
rain on the next day that these same conditions appear?
Properties of Probability Values (Rules of Probability)
11) Real number property, (P(E) = 0? P(E) = 1?)
12) Probability of the sample space S (sum of the probabilities of all
outcomes in S). Examples:
12.1) A sample space consists of four simple outcomes: S = {E, F, G, H}.
If P(E) = 2 5 , P(F) = 1 6 , and P(H) = 3 10 , find P(G).
12.2) Suppose that in a particular group of people, the probability that a
person opposes abortion is five times the probability that a person
favors abortion and the probability that a person is indifferent about
abortion is 0.4 less than the probability that a person opposes
abortion. Find each of the three probability values.
12.3) The probability that a college student votes for the incumbent during
Student Government elections is 0.09, the probability that the student
votes for the lone challenger is 0.07. If the only possible choices are:
to vote for the incumbent, to vote for the lone challenger, or to not
vote at all, find the probability that a college student does not vote
during Student Government elections.
13) Probability of the complement of an event E. (Complement Rule)
13.1) If the probability of rain during a particular day is 1 3 , what is
the probability of no rain on that same day?
13.2) If the probability of drawing an ace from a well-shuffled
standard deck of playing cards is 113 , use the complement rule
to find the probability of not drawing an ace.
13.3) Suppose that the probability that an elevator operates properly at
any time is 0.995. Find the probability that this elevator does not
operate properly at any time.
14) Odds and probability.
14.1) Roll a fair six-sided die once.
a) Find the odds in favor of getting a number less than 3.
b) Find the odds against getting the number 6.
14.2) Draw one card from a well-shuffled deck.
a) Find the odds in favor of getting a Diamond.
b) Find the odds against getting an Ace.
14.3) If the probability that a candidate wins an election is 0.35,
a) Find the odds in favor of this candidate winning the election.
b) Find the odds against this candidate winning.
14.4) If the probability of rain on a particular day is 0.42,
a) Find the odds in favor of rain on that day.
b) Find the odds against rain on that day.
15) Converting odds to probability.
15.1) Suppose 7:2 are the odds in favor of Pearl winning a horse race.
a) Find the probability that Pearl wins the race.
b) Find the probability that Pearl does not win the race.
15.2) Suppose 3:7 are the odds against a candidate winning an election.
a) Find the probability that the candidate wins the election.
b) Find the probability that the candidate does not win the election.
16) Addition Rule of Probability.
16.1) For disjoint events.
16.2) General Addition Rule.
17) Suppose S = {a, b, c, d, e, f, g, h, i, j} is the sample space of an experiment
and E = {a, c, d}, F = {b, e, f, g}, G = {b, c, d, h, j} are events from S.
a) Are E, F disjoint? b) Are E, G mutually exclusive?
c) Are F and G disjoint?
18) In each case, decide whether the given events are mutually exclusive.
a) E = {a, c, x}; F = {b, d}
b) E = {t, e, a, m}; F = {s, e, a}
c) E: Being Male; F: Having Cervical Cancer
d) E: Being a Politician; F: Being an Actor.
e) E: Owning 50 Helicopters; F: Owning 60 Speed-Boats
19) Let E represent the event “getting Heads” and F represent the event
“getting Tails” when tossing a coin once. Are E, F disjoint events?
20) Find the probability of getting an Ace or a Queen when drawing one
card from a well-shuffled deck.
21) Find the probability of getting a number less than 3 or greater than 4
when rolling a fair six-sided die once.
22) In a set of 16 people, 6 are Democrats, 5 are Republicans, 2 are Independents,
and the rest are Reformists. If one person is selected at random from
this set, find the probability that s/he is an Independent or a Reformist.
23) At a particular store, 86% of the shoplifters are caught by the security
officers, 80% are caught by the store closed circuit TV system, and
71% by both. Find the probability that a shoplifter is caught.
24) One card is drawn at random from a well-shuffled deck. Find the
probability that the card is:
a) a diamond or a face card;
b) a red card or an ace;
c) a black card or a face card.
25) A balanced die is rolled once. If E represents the event “getting an even
number” and F represents the event “getting a number less than 5”,
a) Are E and F mutually exclusive?
b) Find the probability of E or F.
c) Find the probability of E and F.
26) At a particular local bank, 60% of the loans are processed by Miss
Alvarez, 55% by Miss Jones, and 43% by both bank officers. One
loan is selected at random from this bank. Find the probability that
the loan was processed by: a) Miss Alvarez or Miss Jones.
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b) Miss Alvarez only.
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c) Neither bank officer.
d) Miss Jones and not by Miss Alvarez.
e) Exactly one of the two bank officers.
f) At least one of the two bank officers.
g) At most one of the two bank officers.
27) Conditional Probability
27.1) One card is drawn from a well-shuffled deck.
(a) If the card is a face card, find the probability that it is a queen.
(b) Find the probability that it is a face card given that it is a queen.
27.2) A particular medicine causes diarrhea to 30% of the people who
use it, a headache to 42%, and causes both side effects to 9% of the
people who use it. (a) Find the probability that a person who uses
this drug experiences a headache if s/he also experiences diarrhea.
(b) If a person who uses this drug experiences diarrhea, find the
probability that s/he experiences a headache also.
27.3) A group of 20 people contains 12 girls and the rest are boys.
Three of the girls suffer from high blood pressure while four of
the boys suffer from high blood pressure. One person is selected
at random from this group. Find the probability that
a) If it is a girl, then she suffers from high blood pressure (HBP)
b) If the person suffers from HBP, then it is a girl
c) the person is a boy given that he suffers from HBP
d) the person suffers from HBP given that he is a boy.
28) Multiplication Rule of Probability
28.1) In a particular region of the world, 90% of the people are
celebrities. Of these, 80% drive luxurious vehicles. One person
is selected at random from this region. Find the probability that
this person is a celebrity and drives a luxurious vehicle.
28.2) Two cards are selected at random without replacement from a
well-shuffled standard deck. (a) Find the probability that both
cards are aces. (b) Find the probability that both are face cards.
(c) Find the probability that the first card is a face card and the
second is an ace. (d) Find the probability that neither card is an ace.
(d) Find the probability of getting a face card and an ace.
28.3) Two cards are selected at random with replacement from a wellshuffled standard deck. (a) Find the probability that both cards are
aces. (b) Find the probability that both are face cards. (c) Find the
probability that the first card is a face card and the second is an ace.
(d) Find the probability of getting a face card and an ace.
28.4) At a particular local bank, 40% of the loans are processed by
Miss Smith. Of these, 5% default. One loan is selected at
random from this bank. Find the probability that the loan is a
default and was processed by Miss Smith.
28.5) At a particular state university, 25% of the students are enrolled
in a business major. Of these, 40% are from Florida. Find the
probability that a randomly selected student from this university
is enrolled in a business major and is from Florida.
28.6) Two cards are drawn at random from a well-shuffled deck.
a) If the cards are drawn without replacement, find the probability
that the second card is an ace, given that the first was also an ace.
b) If the cards are drawn without replacement, find the probability
that the second card is an ace, given that the first was a face card.
c) If the cards are drawn with replacement, find the probability
that the second card is an ace, given that the first was also an ace.
d) If the cards are drawn with replacement, find the probability
that the second card is an ace, given that the first was a face card.
29) INDEPENDENT EVENTS
29.1) At a certain local bank, 40% of the loans are processed by Miss
Smith, 30% by Miss Alvarez, and 12% by both bank officers.
Are the events “processed by Miss Smith” and “processed by
Miss Alvarez” statistically independent?
29.2) In some region of the world, 90% of the people are celebrities,
80% drive a luxurious vehicle, and 70% are celebrities and drive
luxurious vehicles. Are the events “being a celebrity” and
“driving a luxurious vehicle” statistically independent?
29.3) Suppose E, F are events from a sample space S. If E, F are
independent events, find P(E ∪ F).
29.4) A building has two elevators that operate independently. If the
probability that an elevator operates properly at any time is 0.99,
find the probability that: a) both elevators operate properly.
b) neither elevator operates properly.
c) elevator 1 operates properly, but not elevator 2.
d) exactly one elevator operates properly.
e) at least one elevator operates properly.
f) at most one elevator operates properly.
29.5) Suppose the probability of passing a CPA examination is 0.3. If
Amy and Mark take the CPA exam, find the probability that
a) both pass; b) both fail; c) Amy passes, but not Mark;
d) exactly one them passes the exam; e) at least one of them
passes the exam; f) at most one of them passes the exam.
30) CONTINGENCY TABLES
30.1) In a group of 250 people, 140 are women (W) and the rest are men
(M). Of the women, 30 enjoy baseball (Bs), 70 enjoy football (F),
and the rest enjoy basketball (Bk). Of the men, 20 enjoy baseball,
50 enjoy football, and the rest enjoy basketball. Construct the joint
frequency distribution of gender versus sports. If one person is
selected at random from this group, find the probability that:
a) the person is a woman; b) the person is basketball fan; c) the
person is a man or a baseball fan; d) the person is a woman and a
football fan; e) the person is a woman given that she is a
basketball fan; f) the person is basketball fan given that s/he is a
woman; g) if the person is a man, s/he is a football fan; h) the
person is baseball if s/he is a man; i) a baseball fan is a woman;
j) a basketball fan is a man; k) P(W′); l) P(W ∪ F); m) P(M ∩ Bk);
n) P(W|F); o) P(Bs|M)
30.2) 400 customers showed up at a car dealership during a particular
weekend. Only 320 made a purchase (M). Of these, 300 were
satisfied with the service (S), 15 were not (D), and the rest were
indifferent about the service they received (I). Of those who did
not make a purchase (N), 70 were satisfied with the service, 9 were
not, and the rest were indifferent about the service. Construct the
contingency table of customers versus satisfaction. If one customer
is selected at random from this group, find the probability that:
a) the person made a purchase; b) the person was dissatisfied
with the service; c) the person did not make a purchase and was
satisfied with the service; d) the person made a purchase or was
indifferent about the service; e) the person made a purchase given
that s/he was dissatisfied; f) the person was satisfied given that
s/he did not make a purchase; g) if the person did not make a
purchase, s/he was indifferent; h) the person made a purchase if
s/he was satisfied; i) an dissatisfied person made a purchase; j) a
person who made a purchase was indifferent; k) P(M′); l) P(N ∪ I);
m) P(M ∩ D); n) P(N|D); o) P(S|M)
31) BAYES’ RULE
Computational Formulas:
a) If E = (E ∩ F) ∪ (E ∩ F ′), then P(F | E ) =
P (F ) ⋅ P (E | F )
P (F ) ⋅ P (E | F ) + P (F ′) ⋅ P (E | F ′)
b) If E = (E ∩ F) ∪ (E ∩ G) ∪ (E ∩ H), then
P (F | E ) =
P (F ) ⋅ P (E | F )
P(F ) ⋅ P(E | F ) + P(G ) ⋅ P(E | G ) + P(H ) ⋅ P(E | H )
n
c) General Bayes’ Formula: If E =  ( E ∩ Fi ) , then
P (Fi | E ) =
P (Fi ) ⋅ P ( E | Fi )
i =1
n
∑ P (F ) ⋅ P (E | F )
i =1
i
i
31.1) In some region of the world, 3% of the people suffer from a
particular illness. A medical diagnosis test has been developed to
determine who suffers from this illness and who doesn’t. The
test comes back positive on 98% of the people who actually have
the disease and the test comes back positive on 4% of the people
who actually do not have the disease. Find:
a) the probability that the test results are positive
b) the probability that the test results are negative
c) the probability that if a person’s test results are positive, then
the person has the disease.
d) the probability that if a person’s tests results are negative, then
the person does not have the disease
e) the probability that if a person’s tests results are positive, then
the person does not have the disease
f) the probability that if a person’s tests results are negative, then
the person does have the disease
31.2) At a particular local bank, 60% of the loans are processed by
Miss Smith and the rest by Miss Alvarez. Five percent of the
loans processed by Miss Smith default, while four percent of the
loans processed by Miss Alvarez default. Find the probability that
a) a loan defaults given that it was processed by Miss Alvarez
b) a defaulted loan was processed by Miss Smith
c) if a loan is good, then it was processed by Miss Alvarez
d) a loan processed by Miss Smith is a good loan.
31.3) A manufacturer of golf balls uses three machines (A, B, C) for
the entire production. Machine A produces 40% of all the golf
balls, machine B produces 35%, and machine C produces the
rest. Three percent of the golf balls from machine A are
defective, four percent of the golf balls from machine B are
defective, and two percent of the golf balls from machine C are
defective. Find the probability that
a) a golf ball is defective
b) if a golf ball is not defective, it was produced by machine C
c) if a golf ball is defective, it was produced by machine B
d) a golf ball is defective given that it was produced by machine A
e) a golf ball was produced by machine A, given that it was defective