Chapter 4

Selecting a student at random from a list of 10,000 students at a large university

Each of the 10,000 students is a possible outcome for this experiment, so the sample space consists of the 10,000 students.

The Empirical Method can be used to calculate the exact probability of an event.

F

notation

If A denotes an event, the probability of the event A is denoted P(A).

General Method for computing conditional probabilities

provides a way to compute probabilities for events of the form "A and B." If we multiply both sides of the equation by P(A) we obtain the General Multiplication Rule.

When sampling with replacement, the sampled items are independent. When sampling without replacement, if the ______________ of the population, the sampled items may be treated as independent. When sampling without replacement, if the_________________ of the population, the sampled items cannot be treated as independent.

sample size is less than 5% sample size is more than 5%

We can replace the first item drawn before sampling the second; this is known as_____________________.

sampling with replacement

The other option is to leave the first item out when sampling the second one; this is known as ____________________. When sampling without replacement, it is impossible to sample an item more than once.

sampling without replacement

Law of Large Numbers

says that as a probability experiment is repeated again and again, the proportion of times that a given event occurs will approach its probability.

Conditional Probability

the probability of an event ( A ), given that another ( B ) has already occurred.

Venn Diagram

to illustrate mutually exclusive events. In a Venn diagram, the sample space is represented by a rectangle, and events are represented by circles drawn inside the rectangle. If two circles do not overlap, the two events cannot both occur. If two circles overlap, the overlap area represents the occurrence of both events.

A probability that is computed with the knowledge of additional information is called a conditional probability; a probability computed without such knowledge is called an _______________. As this example shows, the ________________ of an event can be much different than the unconditional probability.

unconditional probability conditional probability

If P(A)= 0.3, P(B) = 0.4, and P(A or B) = 0.7, are A and B mutually exclusive?

yes

A pollster plans to sample 1500 voters from a city in which there are 1 million voters. Can the sampled voters be treated as independent? Explain.

yes because the sample size is less than 5%

Or Are the mutually exclusive Yes

P(A or B )=P(A)+P(B)

compound event

P(A or B) = P(A occurs or B occurs or both occur)

Addition Rule for Mutually Exclusive Events:

P(A or B) = P(A) + P(B)

addition rule for mutually exclusive events

P(A or B) = P(A) + P(B)

The General Addition Rule

P(A or B) = P(A) + P(B) - P(A and B)

General Addition Rule

P(A or B) = P(A) + P(B) − P(A and B)

Rule of Complements

P(Ac) = 1 - P(A)

Rule of Complements:

P(Ac) = 1 − P(A)

At least

P(At least one )=1-P(none)

If P(A) = 0.35, find P(Ac).

.65

According to recent figures from the U.S. Census Bureau, the percentage of people under the age of 18 was 23.5% in New York City, 25.8% in Chicago, and 26.0% in Los Angeles. If one person is selected from each city, what is the probability that all of them are under 18? Is this an unusual event?

0.0158

Five lifeguards are available for duty one Saturday afternoon. There are three lifeguard stations. In how many ways can three lifeguards be chosen and ordered among the stations?

60

Ten runners enter a race. The first-place finisher will win a gold medal, the second-place finisher will win a silver medal, and the third-place finisher will win a bronze medal. In how many different ways can the medals be awarded?

720

Fundamental Principle of Counting

A method for counting the number of outcomes (the size of the sample space) of a probabilistic situation, often in which the order of the items matters.

The General Addition Rule is used for probabilities of the form P(A or B).

T

The _________________of an event B, given an event A, is denoted P(B | A). -P(B | A) is the probability that B occurs, under the assumption that A occurs. -We read P(B | A) as "the probability of B, given A."

conditional probability

empirical method

consists of repeating an experiment a large number of times, and using the proportion of times an outcome occurs to approximate the probability of the outcome.

sample space

contains all the possible outcomes of a probability experiment.

If a sample space has n __________________, and an event A has k outcomes, then

equally likely outcomes

probability model

for a probability experiment consists of a sample space, along with a probability for each event.

In the case of two coin tosses, the outcome of the first toss does not affect the second toss. Events with this property are said to be ________________.

independent

Two events are said to be___________________ if it is impossible for both events to occur.

mutually exclusive

Combination Formula

n!/(n-r)!r!

The number of combinations of r objects chosen from n is often denoted by the symbol nCr. The reasoning above can be generalized to derive an expression for____________.

nCr

Five hundred students attend a college basketball game. Fifty of them are chosen at random to receive a free T-shirt. Can the sampled students be treated as independent? Explain.

no because the sample size is more than 5%

Permutation Formula

of r items chosen from n items is an ordering of the r items. It is obtained by choosing r items from a group of n items, then choosing an order for the r items.

Equally likely outcomes

outcomes that have the same chance of occurring

The word_____________is another word for ordering. When we count the number of permutations, we are counting the number of different ways that a group of items can be ordered.

permutation

Among those who apply for a particular job, the probability of being granted an interview is 0.1. Among those interviewed, the probability of being offered a job is 0.25. Find the probability that an applicant is offered a job.

0.025

If P(A) = 0.2, P(B) = 0.5, and A and B are mutually exclusive, find P(A or B).

0.7

If P(A) = 0.75, P(B) = 0.4, and P(A and B) = 0.25, find P(A or B).

0.9

A fair coin is tossed five times. What is the probability that it comes up heads at least once?

1-(1/2)^5

Items are inspected for flaws by three inspectors. If a flaw is present, each inspector will detect it with probability 0.8. The inspectors work independently. If an item has a flaw, what is the probability that at least one inspector detects it?

1-0.8 1-(.2)^3

Five runners run a race. One of them will finish first, another will finish second, and so on. In how many different orders can they finish?

120

Of the 30 people in attendance, 12 are men and 18 are women. What is the probability that all the prize winners are men? What is the probability that at least one prize winner is a woman?

792 0.9944

probability experiment

A probability experiment is one in which we do not know what any individual outcome will be, but we do know how a long series of repetitions will come out.

A)More than 50 of them are business majors. B)At least 50 of them are business majors. C)Fewer than 50 of them are business majors. D)Exactly 50 of them are business majors.

A)If it is not true that more than 50 are business majors, then the number of business majors must be 50 or less than 50. The complement is that 50 or fewer of the students are business majors. B)If it is not true that at least 50 are business majors, then the number of business majors must be less than 50. The complement is that fewer than 50 of the students are business majors. C)If it is not true that fewer than 50 are business majors, then the number of business majors must be 50 or more than 50. Another way of saying this is that at least 50 of the students are business majors. The complement is that at least 50 of the students are business majors. D)If it is not true that exactly 50 are business majors, then the number of business majors must not equal 50. The complement is that the number of business majors is not equal to 50.

___________________is an outcome or a collection of outcomes from a sample space.

An event

For any positive integer n, the number n! is pronounced "n factorial" and is equal to the product of all the integers from n down to 1. By definition, 0!=1.

Counting the number of permutations

The Fundamental Principle of Counting

If an operation can be performed in m ways, and a second operation can be performed in n ways, then the total number of ways to perform the sequence of two operations is mn.

Does one event prevent the other from occurring? no does the occurrence of one event change the probability of another no

Independent

Does one event prevent the other from occurring? Yes

Mutually exclusive

Does one event prevent the other from occurring? no does the occurance of one event change the probability of another yes

Neither

A: Sophie is a member of the debate team; B: Sophie is the president of the theater club.

Not Mutally exclusive

Does the event involve and (independent) NO

P( A and B)=P(A)P(B/A)

If events A and B are mutually exclusive, then_______________. This leads to a simplification of the General Addition Rule.

P(A and B) = 0

General Multiplication Rule

P(A and B) = P(A) * P(B|A)

General Multiplication Rule

P(A and B) = P(A)P(B | A)= P(B)P(A | B)

Multiplication Rule for Independent Events

P(A and B) = P(A)P(B)

Multiplication Rule for Independent Events:

P(A and B) = P(A)P(B)

Does the event involve and (independent) Yes

P(A and B) =P(A)P(B)

addition rule for mutually exclusive events

P(A or Ac) = P(A)+P(Ac).

Or Are the mutually exclusive No

P(A or B)=P(A)+P(B)-P(A and B)

__________of an event is the proportion of times that the event occurs in the long run. So, for a "fair" coin, that is, one that is equally likely to come up heads as tails, the probability of heads is 1/2 and the probability of tails is 1/2.

Probability

Thirty people attend a certain event, and 5 will be chosen at random to receive prizes. The prizes are all the same, so the order in which the people are chosen does not matter. How many different groups of 5 people can be chosen?

Since the order of the 5 chosen people does not matter, we need to compute the number of combinations of 5 chosen from 30. This is 142,506

When sampling with replacement, it is possible to draw the same item more than once. (T/F)

T

notation

The number of permutations of r items chosen from n is denoted nPr.

Conditional Probabilities

The probability of one event given the known outcome of a (possibly) related event.

There are six possible outcomes for the roll of a die: the numbers from 1 to 6. So a sample space is {1, 2, 3, 4, 5, 6}.

The roll of a die

License plates in a certain state contain three letters followed by three digits. How many different license plates can be made

There are six operations in all: choosing three letters and choosing three digits. There are 26 ways to choose each letter and 10 ways to choose each digit. The total number of license plates is therefore

The toss of a coin

There are two possible outcomes for the toss of a coin: Heads and Tails. So a sample space is {Heads, Tails}.

A die is rolled. Event A is that the die comes up 3, and event B is that the die comes up an even number.

These events are mutually exclusive. The die cannot both come up 3 and come up an even number.

Determine whether the following pairs of events are independent: A college student is chosen at random. The events are "being a freshman" and "being less than 20 years old." A college student is chosen at random. The events are "born on a Sunday" and "taking a statistics class."

These events are not independent. If the student is a freshman, the probability that the student is less than 20 years old is greater than for a student who is not a freshman. These events are independent. If a student was born on a Sunday, this has no effect on the probability that the student takes a statistics class.

A fair coin is tossed twice. Event A is that one of the tosses is a head, and event B is that one of the tosses is a tail.

These events are not mutually exclusive. If the two tosses result in HT or TH, then both events occur.

counting

When computing probabilities, it is sometimes necessary to count the number of outcomes in a sample space without being able to list them all. In this section, we will describe several methods for doing this.

Sometimes we need to find the probability that an event occurs _______________once in several independent trial

at least

For example, we may not care which lifeguard occupies which station; we might care only which three lifeguards are chosen. Each distinct group of objects that can be selected, without regard to order, is called a ________________.

combination

If A is any event, the _________________ of A is the event that A does not occur. The complement of A is denoted Ac.

complement

If there is a 60% chance of rain today, then there is a 40% chance that it will not rain. The events "Rain" and "No rain" are complements. The complement of an event A is the event that A does not occur.

complement

If you go out in the evening, you might go to dinner, or to a movie, or to both dinner and a movie. In probability terminology, "go to dinner and a movie" and "go to dinner or a movie" are referred to as compound events, because they are composed of combinations of other events—in this case the events "go to dinner" and "go to a movie."

compound event

_____________is an event that is formed by combining two or more events.

compound event