Math 123- Statistics
Chapter 3 Notes
Name_______________________________
3.1 Basic Concepts of Probability
The most important thing to know how to do in Chapter 3 is write the probability
notation. You will be required to use this notation in your homework.
Def- A probability experiment is a trial through which specific results are obtained.
Def- The result of a single trial in a probability experiment is called an outcome.
Def- The set of all outcomes is called the sample space.
Def- An event consists of one or more outcomes and is a subset of the sample space.
Def- An event that consists of a single outcome is called a simple event.
Ex- a) Identify the sample space of tossing a coin four times.
b) Identify the outcome(s) of the event “getting three heads.”
Ex- State the sample space for the months of the year.
Ex- State the sample space for the years since you were born.
Ex- State the sample space for the names of your full-blooded siblings.
Def- Classical probability or theoretical probability is used when each outcome in a sample space is
equally likely to occur.
P(E)=
Def- Empirical probability or statistical probability is based on observations obtained from probability
experiments. The empirical probability of an event E is the relative frequency of the event E.
P(E)=
Def- Subjective probability results from guesses and estimates.
Ex- Classify as classical, empirical, or subjective probability.
a) Based on prior counts, a quality control officer says there is a .05 probability that a randomly
chosen part is defective.
b) The probability of randomly selecting 5 cards of the same suit from a standard deck is .002.
c) A stock market analyst predicts that the chance that Corporation A’s stock price will fall today is
75%.
Law of Large Numbers- As an experiment is repeated over and over, the experimental probability of
an event approaches the actual probability of the event.
Def- The complement of an event E is the set of all outcomes in the sample space not including E.
Notation: E’= complement of E
Ex- The U.S. age distribution for the 2000 census is shown below.
Age
At most 19 20 – 34 35 – 59 60 – 84 At least 85
Population 29%
21%
34%
15%
1%
a) What is the probability that a person is at least 20 years old?
b) What is the probability that a person will be less than 60 years old?
c) Find the probability that a person is at most 84 years old.
Range of Probability Rule
The probability of an event E is between 0 and 1, inclusive.
0  P( E )  1
Fundamental Counting Principle
If one event can occur in m ways and a second event can occur in n ways, then the number of ways
the two events can occur in sequence is mn. This rule can be extended to any number of events
occurring in sequence.
Ex- If a lunch consists of a drink, a sandwich, and a cookie, find the number of lunches possible if
there are 4 drink choices, 7 sandwich choices, and 3 cookie choices.
Ex- How many 5 letter words can you make with the alphabet assuming that letters can be
repeated?
3.2 Conditional Probability and the Multiplication Rule
Def- A conditional probability is the probability of an event occurring given that another event has
already occurred. The probability of B occurring giving that A has occurred is denoted PB A.
Ex- For each of the following, use the conditional probability notation to express the sentence
mathematically.
a) The probability that a person likes ice-cream given that they like cookies is .83.
b) Of all students who receive one or more plus grades (D+, C+, B+, A+), 92% were
undergraduates.
c) Of all the animals in the forest, 22% are bears.
Def- Two events are independent if the occurrence of one of the events does not affect the
probability of the occurrence of the other event. Two events A and B are independent if
PB A  P(B) or if PA B   P(A) .
Def- Two events that are not independent are dependent.
The Multiplication Rule for the Probability of A and B
The probability that any
two events A and B will occur in sequence is
P( AandB)
.)
P( AandB)  P( A) PB A . (I personally prefer to think of the formula as PB A 
P( A)
The probability that two
P( AandB)  P( A) PB .
independent events A and B will occur in sequence is
Ex- Determine if, when selecting from a standard deck of cards, the events “getting a king” and
“getting a heart” are independent events.
Ex- The graduation rate for Righetti High School is 94%. Find the probability that in a group of five
randomly selected Righetti High School students, the first four graduate, and the fifth student does
not graduate.
Ex- A sock drawer has nine pairs of socks with three pairs each of white, black, and blue socks.
What is the probability that you will select a black pair, remove it, and then select either a blue or a
white pair?
Ex- The probability that a person in the U.S. has O+ blood is 38%. Three people are selected at
random.
a) Find the probability that all three people have O+ blood.
b) Find the probability that none of the three people chosen have O+ blood.
c) Find the probability that at least one of the people chosen has O+ blood.
d) Find the probability that the second person has O+ blood, but the first person and third person do
not.
3.3 The Addition Rule
Def- Two events are mutually exclusive if A and B can’t occur at the same time.
The Addition Rule for the Probability of A or B
P( AorB)  P( A)  P( B)  P( AandB)
Ex- A random sample of 250 working adults found that 37% access the internet at work, 44%
access the internet at home, and 21% access the internet at both work and at home. What is the
probability that a person in this sample selected at random accesses the internet at home or at
work?
Ex- A card is randomly selected from a standard deck of cards. Find the probability that the card is
between 4 and 8 (inclusive) or is a club.
Ex- A twelve-sided die has sides numbered 1 – 12.
a) Find the probability that the roll results in an odd number or a number less than four.
b) Find the probability that the roll results in a multiple of three or the number six.
Ex- Sixty-nine percent of adults are enrolled in college. Eleven percent of adults are employed.
a) Find the proportion of adults that are enrolled in college and employed if 75% of adults are
enrolled in college or employed.
b) Are the events “enrolled in college” and “employed” independent events?
Ex- Six percent of adults are enrolled in college. Of those adults 55% of them are employed. Find
the proportion of adults that are enrolled in college and employed.
3.4 Counting Principles
Def- A permutation is an ordered arrangement of objects. The number of permutations of n objects
is n!.
Ex- Find the number of ways to arrange 6 chairs in a row.
Ex- Find the number of ways to arrange 12 books on one shelf of a bookshelf.
Formula- The number of permutations of n objects taken r at a time is n Pr 
n!
for r  n .
(n  r )!
Ex- Fifteen cyclists enter a race. In how many ways can the cyclists finish 1st, 2nd, 3rd?
Ex- 6241 people buy raffle tickets. In how many ways can 6 cash awards, each of a different
monetary value, be passed out?
Formula- The number of distinguishable permutations for n objects where n1 are of one type, n 2 are
n!
of another type, etc. is
where n  n1  n2  ...  nk .
n1! n 2 !...n k !
Ex- How many distinguishable words can you get from AAABBBCCCC?
Ex- How many distinguishable ways can you order 5 red balls, 3 green balls, and 4 blue balls?
Def- A combination is a selection of r objects from a group of n objects without regard to order and is
n!
denoted n C r where n C r 
.
(n  r )! r!
Ex- A florist has 12 different flowers from which flower arrangements can be made. If a centerpiece
is to be made using 5 different flowers, how many different centerpieces can be made?
Ex- A total of 350 plants at a local nursery were tested to see if they have root decay. Seventeen of
those plants were found to have root decay. Find the probability that in a sample of six randomly
selected plants from the nursery, two will have root decay.
Ex- A batch of 200 calculators contains 10 defective calculators.
a) What is the probability that if 3 calculators are randomly selected, none of them will have defects?
b) What is the probability that if 3 calculators are randomly selected, all have defects?
c) What is the probability that if 3 calculators are randomly selected, at least one will have a defect?
d) What is the probability that if 18 calculators are randomly selected, 11 of them will not have
defects?