Chapter 9

Counting, Probability Distribution and Further

Topics on Probability

9.1 Probability Distributions and Expected Value

A random variable is a function that assigns a real number to each outcome of an experiment.

Example: “number of heads” showing when 2 coins are tossed•Probability distributions

Table with all possible values of random variables, together with the corresponding probabilities.

•Histogram: bar graph used to display probability distributions.

Histogram

0.13

0.29

0.38

0.13

0.08

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

1 2 3 4 5

X 1 2 3 4 5

P(x) .13 .29 .38 .13 .08

P(x 4) = P(x=4) + P(x=5) = .13 + .08 = .21

Examples

1) Give the probability distribution for the number of heads showing when 2 coins are tossed, draw the histogram, and find the probability that at least one coin comes up heads .

Examples

2) Give the probability distribution for the following experiment: 2 dice are rolled, the total number of points are recorded. Draw the histogram, and find the probability that the sum is at least 10.

Examples

3) Give the probability distribution for the following experiment: 3 cards are drawn from a deck, and number of faces is recorded. Draw the histogram, and find the probability that number of faces is 2.

EXPECTED VALUE

Suppose the random variable x can take on the n values x1, x2, …, xn. Also, suppose the probabilities that these values occur are respectively p1, p2, …, pn. Then the expected value of the random variable is:

E(x) = x1p1 + x2p2 + … + xnpn

Example

X 1 2 3 4 5

P(x) .13 .29 .38 .13 .08

Find the expected value of x in the probability distribution below:

Examples

Example 3 (p. 513)

A local church decides to raise money by raffling off a micro oven worth $400, a dinner for two worth $80, and 2 books worth $20. A total of 2000 tickets are sold at $1 each. Find the expected value of winnings for a person who buys 1 ticket in the raffle.

Examples

Example 4 (p. 514)

Hoa and Nam toss a coin to see who buys the coffe ($1.25 a cup). Hoa tosses and Nam calls the outcome. If Nam is correct, Hoa buys the coffee; otherwise Nam pays. Find expected winnings of Hoa.

Examples

Example 5 (p.515)Bob is 50 years old. He must decide 1 of 2 options from an insurance company: get $60000 at age 60 or $65000 at age 70.Given that probability of person living from age 50 to 60 is .88, and probability of person living from age 50 to 70 is .64. Use expected value to help Bob to select which option is more valuable.

Example 7 (p. 516)

Suppose a family has 3 children.

a) Find the probability distribution for the number of girls

b) Use the distribution from part a) to find the expected number of girls in a 3-child family.

Examples

9.2 The Multiplication Principle, Permutations and Combinations

MULTIPLICATION PRINCIPLE• Suppose n choices must be made, with

m1 ways to make choice 1,and for each of these,

m2 ways to make choice 2,and so on, with

mn ways to make choice n.• Then there are m1m2…mn

different ways to make the entire sequence of choices.

Examples

Example 1 (p. 547) Angela has 9 skirts, 8 blouses and 13 different pairs of shoes. How many different skirt-blouse-shoe choices does she have?

Example 3 (p. 547) A combination lock can be set to open to any 3-letter sequence.a) How many sequences are possible?

b) How many sequences are possible if no letter is repeated?

n-FACTORIAL

For any natural number n, we define:

n! = n(n-1)(n-2)…(3)(2)(1).

0! is defined to be the number 1.

Example:

1) Evaluate

2) Evaluate

!9

!10

!3!2

!5

PERMUTATIONDefinition:• A permutation of n elements is an ordering of the

elements.

The number of permutations of an n-element set is n!

• A permutation of r elements (r1) from a set of n elements is an ordering of the r elements.

Number Of Permutations P(n,r):

• If P(n, r) , r n, is the number of permutations of

n elements taken r at a time, then

P(n, r) = )!(

!

rn

n

Examples

• Example 8 (p. 550). A teacher has 5 books and wants to display 3 of them side by side on her desk. How many arrangements of 3 books are possible?

Examples• Example 9 (p. 551): In 2004, 8 candidates

sought for the Democratic nomination for president. In how many ways could voters rank their first, second and third choice?

• Example 10 (p. 551): A televised talk show will include 4 women and 3 men as panelists.

a) In how many ways can the panelists be seated in a row of 7 chairs?

b) In how many ways can the panelists be seated if the men and women are to be alternated?

c) In how many ways can the panelists be seated if men must sit together and the women must sit together?

COMBINATIONDefinition:

• A subset of items selected without regard to order is called combination.

Number Of Combinations:

• If C(n, r) , r n, is the number of

combinations of n elements taken r at a

time, then C(n, r) = )!(!

!

rnr

n

Examples

Example 11 (p. 553):

From a group of 10 students, a committee is to be chosen to meet with the dean. How many different 3-person committee are possible?

Example 12 (p. 553)

In how many ways can a 12-person jury be chosen from a pool of 40 people?

Examples

Example 13 (p. 553)Three managers are to be selected from a group

of 30 to work on a special project.

(a) In how many different ways can the managers se selected?

(b) In how many different ways can the group of 3 be selected if a certain manager must work on the project?

(c) In how many different ways can a nonempty group of at most 3 managers be selected from these 30 managers?

ExamplesExample 14 (p. 554): permutation or combination?

(a)How many 4-digit code numbers are possible if no digits are repeated?

(b) A sample of 3 lightbulbs is randomly selected from a batch of 15 bulbs. How many different samples are possible?

(c) In a basketball conference with 8 teams, how many games must be played so that each team plays every other team exactly once?

(d)In how many ways can 4 patients be assigned to 6 hospital rooms so that each patient has a private room?

ExamplesExample 15 (p. 555)

A manager must select 4 employees for promotion. 12 are eligible.

(a) In how many ways can the 4 be chosen?

(b) In how many ways can 4 employees be chosen to be placed in 4 different jobs?

Example 17

Five cards are dealt from a standard 52-card deck.

(a)How many hands have all face cards?

(b)How many hands have have exactly 2 hearts?

(c)How many 5-card hands have all cards of a single suit?

9.3 APPLICATIONS OF COUNTING

Example 1 (p. 535) From a box containing 1 red, 3 white and 2 green marbles, two marbles are drawn one at a time without replacing the first before the second is drawn. Find the probability that one white and one green marble are drawn.

9.3 APPLICATIONS OF COUNTING

Example 2 (p. 535).

1) How many ways can we select 4 students in a class of 15 students?

2) One student in the class is Nam. Find the probability that Nam will be among the 4 selected students.

3) Find the probability that Nam will not be selected.

9.3 APPLICATIONS OF COUNTING

Example 2 (p. 535).

1) The ski club has 8 women and 7 men. What is the probability that if the club selects 3 officers, all 3 of them will be women?

9.3 APPLICATIONS OF COUNTINGExample 3 (p. 536).

A container has 12 items, in which there are 2 defective. We test 3 items at random; if any of the 3 is defective, the container will not be shipped until each item is checked. Find the probability that this container will not be shipped.

9.3 APPLICATIONS OF COUNTING

Example 4 (p. 537).5 cards are drawn from a

deck. Find:1) P(heart flush)2) P(any flush)3) P(full house of aces and

kings)4) P(any full house)5) P(4 kings)

9.3 APPLICATIONS OF COUNTING

• Example 5 (p. 538)A cooler contains 8 different kinds of soda, among which 3 cans are Pepsi, Classic coke and Sprite. What is the probability, when picking at random, of selecting the 3 cans in the particular order listed in the previous sentence?

• Example 6 (p. 538)There are 5 people in a room. Find the probability that at least 2 of the people have the same birthday.

9.4 BINOMIAL PROBABILITY

BINOMIAL EXPERIMENT

• The same experiment is repeated a fixed number of times.

• There are only two possible outcomes, success or failure.

• The probability of success for each trial is constant.

• The repeated trials are independent.

Examples

• Find the probability of getting 5 ones on 5 rolls of a die.

• Find the probability of getting 4 ones on 5 rolls of a die.

• Find the probability of getting 3 ones on 5 rolls of a die.

BINOMIAL PROBABILITY

If p is the probability of success in a single trial of a binomial experiment, the probability of x successes and n–x failures in n independent repeated trials of the experiment is:

C(n, x) px (1 – p)n–x

Example 1 (p. 544)

In 2003 Supreme Court Justice Sandra announced that she would retire the court. Suppose that 30% of Americans want the new justice to be more liberal, and a random sample of 6 people is chosen, with all responses are independent from each other.

(a)Find the probability that exactly 4 of the people prefer a more liberal justice.

(b) Find the probability that none of the 6 people prefers a more liberal justice.

Example 2 (p. 544 –545)

A family has 3 chidren.

(a) Find the probability distribution for the number of girls.

(b) Find the expected number of girls in a 3-child family.

Expected Value for Binomial Probability

When an experiment meets the four conditions of a binomial experiment with n fixed trials and constant probability of success p, the expected value is:

E(x) = np

Example 3 (p. 545 –546)

Men’s Health magazine reported that 1/3 of men in Boston had significantly low vitamin D levels. Suppose a researcher randomly selects 15 men in Boston.

(a)Find the probability that 6 of the men have low vitamin D levels.

(b) Find the probability that at most 3 of the men have low vitamin D levels.

Example 4 (p. 546)

A survey in China, as reported by the Los Angeles Times, found that 66% of young people feel themselves to be under heavy pressure. If a sample of 8 young people in China is selected, what is the probability that at least 1 will report being under heavy pressure?

Example 5 (p. 546)

If each member of a 9-person jury acts independently of the other members and makes the correct determination of guilty or innocence with probability .65, find the probability that the majority of jurors will reach a correct verdict.

P(x 5) = ?

9. 4 MARKOV CHAINS

MARKOV CHAIN

• One kind of stochastic process, in which the outcome of an experiment depends only on the outcome of the previous experiment.

• Example 1 (p. 549)

• Transition diagram

• Transition matrix

Transition diagram

A B

0.2

0.8

0.35

0.65

65.035.0

2.08.0

B

A

BA

40% 60%

Transition matrixA transition matrix has the following

features:1. It is square, since all possible states

must be used both as rows and as columns.

2.All entries are between 0 and 1, inclusive, because all entries represent probabilities.

3.The sum of the entries in any row must be 1, because the numbers in the row give the probability of changing from the state at the left to one of the states indicated across the top.

65.035.0

2.08.0

B

A

BA

Regular Transition MatrixA transition matrix is regular if some power of the matrix contains all positive numbers.Example:

regularisA

04.6.

5.5.0

025.75.

regularnotisB

100

010

5.05.

A Markov chain is a regular Markov chain if its transition matrix is regular

Probability Vector

• Example 2 (p. 476)

• Probability vector: matrix of only one row, having nonnegative entries with the sum of the entries equal to 1.

• Example of probability vectors: – [0.40, 0.60]– [0.53, 0.47]– [0.40, 0.25, 0.35]

Equilibrium VectorSuppose a regular Markov chain has a transition matrix P.

1. For any initial probability vector v, the products vPn approach a unique vector V as n gets larger and larger. Vector V is called the equilibrium or fixed vector.

2. Vector V has the property that VP = V.

3. Vector V is found by solving a system of equations obtained from the matrix equation VP = V and from the fact that the sum of the entries of V is 1.