Section 7.2 Deﬁnition of Probabilitymayaj/Chapter7_Sec7.2_7... · 2,···,s n} is the sample space of a given experiment, then the probability distribution is a table where the

Section 7.2 Definition of Probability

Question: Suppose we have an experiment that consists of flipping a fair 2-sided coin and observing

if the coin lands on heads or tails? From section 7.1 we should know that there are

outcomes, and that only of those outcomes is a head, but what is the probability of

the coin landing on heads?

Answer:

Probability of heads =# of outcomes with a head

# of total outcomes

Probability of an Event E: Part I If S is the sample place of an experiment with event E, then

the probability of event E occuring, written as P (E), is given by

P (E) =n(E)

n(S)

1. A family has three children. Assuming a boy is as likely as a girl to have been born, what are the

following probabilities?

(a) Three are boys and none are girls.

(b) At least 2 are girls.

2. A pair of fair 6-sided dice is rolled. What is the probability of each of the following? (Round your

answers to three decimal places.)

(a) The sum of the numbers shown uppermost is less than 6

(b) At least one 3 is cast

3. In a sweepstakes sponsored by Gemini Paper Products, 10, 000 entries have been received. If 1

grand prize is drawn, and 4 first prizes, 30 second prizes, and 550 third prizes are to be awarded,

what is the probability that a person who has submitted one entry will win the following?

(a) a third prize (Round your answer to four decimal places if necessary.)

(b) any prize (Round your answer to four decimal places if necessary.)

2 Fall 2017, Maya Johnson

4. If a card is drawn at random from a standard 52-card deck, what is the probability that the card

drawn is one of the following?

(a) a club

(b) a black card

(c) a king

5. In an attempt to study the leading causes of airline crashes, the following data were compiled

from records of airline crashes (excluding sabotage and military action):

Primary Factor Accidents

Flight crew 323

Airplane 47

Maintenance 12

Weather 25

Airport/air tra�c control 25

Miscellaneous/other 14

Assume that you have just learned of an airline crash and that the data give a generally good

indication of the causes of airplane crashes. Give an estimate of the probability that the primary

cause of the crash was due to flight crew or bad weather. (Round your answer to three decimal

places.)


Probability Distribution If S = {s1, s2, · · · , sn} is the sample space of a given experiment,

then the probability distribution is a table where the entries in the first row are all the outcomes

of S and the entries in the second row are all their corresponding probabilities.

Outcomes s1 s2 · · · sn

Probability P (s1) P (s2) · · · P (sn)

6. The grade distribution for a certain class is shown in the following table. Find the probability

distribution associated with these data. (Enter your answers to three decimal places.)

Grade A B C D F

Frequency of

Occurence 5 10 16 7 2

Probability

7. In a survey conducted by a business-advisory firm of 4980 adults 18 years old and older in June

2009, during the ”Great Recession,” the following question was asked: How long do you think

it will take to recover your personal net worth? The results of the survey follow. (Round your

answers to three decimal places.)

Answer (years) 1� 2 3� 4 5� 10 � 10

Respondents 1004 1301 2114 561

(a) Determine the empirical probability distribution associated with these data.

Answer (years) 1� 2 3� 4 5� 10 � 10

Probability

(b) If a person who participated in the survey is selected at random, what is the probability that

he or she expects that it will take 5 or more years to recover his or her personal net worth?


Properties of Probability Distributions

1. 0 P (si) 1 for i = 1, 2, · · · , n.

2. P (s1) + P (s2) + · · ·+ P (sn) = 1

3. P ({si} [ {si}) = P (si) + P (si) (i 6= j) for i = 1, 2, · · · , n; j = 1, 2, · · · , n.

Finding the Probability of an Event E: Part II If E = {s1, s2, · · · , sm} is an event of a sample

space S of a given experiment, then

P (E) = P (s1) + P (s2) + · · ·+ P (sm)

If E is the empty set, ?, then P (E) = 0.

8. A die is loaded, and it has been determined that the probability distribution associated with

the experiment of rolling the die and observing which number falls uppermost is given by the

following:

Simple Event Probability

{1} .18

{2} .13

{3} .19

{4} .2

{5} .15

{6} .15

(a) What is the probability of the number being even?

(b) What is the probability of the number being either a 1 or a 6?

(c) What is the probability of the number being less than 4?


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9. Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with the experiment having the

following probability distribution. (Enter your answers as fractions.)

Outcomes s1 s2 s3 s4 s5 s6

Probability 112

112

312

112

412

212

(a) Find the probability of A = {s1, s3}.

(b) Find the probability of B = {s2, s4, s5, s6}.

(c) Find the probability of C = S.

10. The sample space S = {s1, s2, s3} has the property that P (s1)+P (s2) = 0.39 and P (s2)+P (s3) =

0.83. Find the probability of each outcome.


Section 7.3 Rules of Probability

Definition: Two events E and F are said to bemutually exclusive if the two events have no outcomes

in common, that is E \ F = ?.

Properties of Probabilities: Revisited

1. 0 P (E) 1 for any event E.

2. P (S) = 1.

3. If E and F are mutually exclusive, then P (E [ F ) = P (E) + P (F ).

4. If E and F are any two events of an experiment, then P (E [ F ) = P (E) + P (F ) � P (E \ F ).

(This should remind us of the union/addition rule from section 6.2.)

5. If E is an event of an experiment and Ec denotes the complement of E, then P (Ec) = 1� P (E).

The reverse is also true, namely, P (E) = 1� P (Ec). (I call this the Complement Rule)

1. An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment

and find the probability of the event

A diamond or a king is drawn.

2. Let E and F be two events of an experiment with sample space S. Suppose P (E) = 0.59,

P (F ) = 0.38, and P (Ec \ F ) = 0.28. Calculate the probabilities below.

(a) P (Ec)

(b) P (E \ F )

(c) P (E [ F )

(d) P (Ec \ F c)

(e) P (E [ F c)


3. Let E and F be two mutually exclusive events, and suppose P (E) = 0.6 and P (F ) = 0.1.

Compute the probabilities below.

(a) P (E \ F )

(b) P (E [ F )

(c) P (Ec)

(d) P (Ec \ F c)

(e) P (Ec [ F c)

4. Let S = {s1, s2, s3, s4, s5, s6} be the sample space associated with an experiment having the

following partial probability distribution.

Outcomes s1 s2 s3 s4 s5 s6

Probability 429

1029

229

329

229

Consider the events: A = {s1, s2, s5}, B = {s3, s5, s6}, C = {s1, s3, s4, s6}, and D = {s1, s2, s3}

Calculate the following probabilities. (Give answers as fractions.)

(a) P (s2)

(b) P (D)

(c) P (Bc)

(d) P (Ac \ B)

(e) P (Cc [D)


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5. Among 500 freshmen pursuing a business degree at a university, 317 are enrolled in an economics

course, 214 are enrolled in a mathematics course, and 138 are enrolled in both an economics and a

mathematics course. What is the probability that a freshman selected at random from this group

is enrolled in each of the following? (Round answers to three decimal places.)

(a) an economics or a mathematics course

(b) exactly one of these two courses

(c) neither an economics course nor a mathematics course

6. The following table gives the percentage of music downloaded from the United States and other

countries by U.S. users:

Country U.S. Germany Canada Italy U.K. France Japan Other

Percent 44.5 16.1 7.1 6.4 4.5 3.4 3.0 15.0

(Round answer to three decimal places.)

(a) Verify that the table does give a probability distribution for the experiment.

(b) What is the probability that a U.S. user who downloads music, selected at random, obtained

it from either the United States or Canada?

(c) What is the probability that a U.S. user who downloads music, selected at random, does not

obtain it from Italy, the United Kingdom (U.K.), or France?


7. A poll was conducted among 250 residents of a certain city regarding tougher gun-control laws.

The results of the poll are shown in the table. (Round answers to three decimal places.)

Own Own Own a

Only a Only a Handgun Own

Handgun Rifle and a Rifle Neither Total

Favor Tougher Laws 0 11 0 139 150

Oppose Tougher Laws 64 4 21 0 89

No Opinion 0 0 0 11 11

Total 64 15 21 150 250

(a) If one of the participants is selected at random, what is the probability that he or she favors

tougher gun-control laws?

(b) If one of the participants is selected at random, what is the probability that he or she owns

a handgun?

(c) If one of the participants is selected at random, what is the probability that he or she owns

a handgun but not a rifle?

(d) If one of the participants is selected at random, what is the probability that he or she favors

tougher gun-control laws and does not own a handgun?


Section 7.4 Use of Counting Techniques in Probability

Question: Five marbles are selected at random without replacement from a jar containing four white

marbles and six blue marbles. From Section 6.4, we know that there are ways to

choose these five marbles. We should also know that of those samples have all blue

marbles. How do we find the probability that our sample has all blue marbles?

Computing the Probability of an Event in a Uniform Sample Space: Revisited

Let S be a uniform sample space, and let E be any event. Then

P (E) =Number of outcomes in E

Number of outcomes in S=

n(E)

n(S)

Let’s revisit the above question:

1. Five marbles are selected at random without replacement from a jar containing four white marbles

and six blue marbles. Find the probability of the given event. (Round answer to three decimal

places.)

All of the marbles are blue.

2. A 4-card hand is drawn from a standard deck of 52 playing cards. Find the probability that the

hand contains the given cards. (Round answer to 3 decimal places.)

No diamonds.

3. Three cards are selected at random without replacement from a well-shu✏ed deck of 52 playing

cards. Find the probability of the given event. (Round answer to four decimal places.)

Three cards of the same suit are drawn.


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4. A box has 7 marbles, 3 of which are white and 4 of which are red. A sample of 4 marbles is

selected randomly from the box without replacement. (Give answers as an exact fraction.)

(a) What is the probability that exactly 2 are white and 2 are red?

(b) What is the probability that at least 2 of the marbles are white?

5. Jacobs & Johnson, an accounting firm, employs 20 accountants, of whom 6 are CPAs. If a

delegation of 4 accountants is randomly selected from the firm to attend a conference, what is

the probability that 4 CPAs will be selected? (Round answer to three decimal places.)

6. A shelf in the Metro Department Store contains 90 colored ink cartridges for a popular ink-jet

printer. Eight of the cartridges are defective. (Round answers to five decimal places.)

(a) If a customer selects 2 cartridges at random from the shelf, what is the probability that both

are defective?


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(b) If a customer selects 2 cartridges at random from the shelf, what is the probability that at

least 1 is defective?

7. A customer from Cavallaro’s Fruit Stand picks a sample of 4 oranges at random from a crate

containing 65 oranges, of which 5 are rotten. What is the probability that the sample contains 1

or more rotten oranges? (Round answer to three decimal places.)

8. A student studying for a vocabulary test knows the meanings of 14 words from a list of 26 words.

If the test contains 10 words from the study list, what is the probability that at least 8 of the

words on the test are words that the student knows? (Round answer to three decimal places.)


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Section 7.2 Deﬁnition of Probabilitymayaj/Chapter7_Sec7.2_7... · 2,···,s n} is the sample space of a given experiment, then the probability distribution is a table where the