11.2 – Probability – Events Involving “Not” and “Or” Properties of Probability 1. The...

11.2 – Probability – Events Involving “Not” and “Or”Properties of Probability

1. The probability of an event is between 0 and 1, inclusive.

2. The probability of an impossible event is 0.

3. The probability of a certain event is 1.

Examples:

Rolling a fair die, what is the probability of each event?

a) The number 3 is rolled. P(3) = 1/6

b) A number not 3 is rolled.

c) The number 9 is rolled.

d) A number < 7 is rolled.

P(not 3) = 5/6

P(9) = 0

P(< 7) = 1

11.2 – Probability – Events Involving “Not” and “Or”

Probability of a Complement

The probability that an event E will not occur is equal to one minus the probability that it will occur.

P(not E) = 1 – P(E)

Other forms of the equation:

What is the probability of not drawing an ace from a standard deck of 52 cards?

P(E) + P(E) = 1

P(E) = 1 – P(E)

P(not an ace) = 1 – P(ace)

P(not an ace) = 48/52 = 12/13

P(not an ace) = 1 – (4/52)

Events Involving “Not”

P(E) = 1 – P(E)


The probability that one event or another event will occur usually involves the union and addition.

Mutually Exclusive Events

Addition Rule of Probability (for A or B)

Mutually exclusive events can not occur simultaneously

If A and B are any two events, then

If A and B are mutually exclusive, then

Events Involving “Or”

Two events, A and B, are mutually exclusive events if they have no outcomes in common.

P(A or B) = P(A) + P(B) – P(A and B)

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


=

1. What is the probability of drawing a king or a diamond from a standard deck of 52 cards?

1/52

4/13

P(king or diamond) =

Examples:

P(K) + P(D) – P(K and D)

4/52 + 13/52 –

= 16/52 =

11.2 – Probability – Events Involving “Not” and “Or”Examples:

=

2. What is the probability of a 2 or an odd number being rolled on a fair die?

Mutually exclusive events

2/3

P(2 or odd) = P(2) + P(odd)

1/6 + 3/6

= 4/6 =

The probability of an event based on the fact that some other event has occurred, will occur, or is occurring.

P(B/A) =

Conditional Probability

The probability of event B occurring given that event A has occurred is usually stated as “the conditional probability of B, given A; P(B/A)

11.3 – Conditional Probability – Events Involving “And”

)(

)(

AP

BAP

)(

)(

AP

BandAP

Example:

A number from the sample space S = {2, 3, 4, 5, 6, 7, 8, 9} is randomly selected. Given the defined events A and B,

A: selected number is odd, and B: selected number is a multiple of 3

find the following probabilities.

a) B = {3, 6, 9}


b) P(A and B) =

a) P(B) b) P(A and B) c) P(B/A)


P(B) = 3/8

P({3, 5, 7, 9} {3, 6, 9})

= P({3, 9}) = 2/8 = 1/4

c) Probability of B given A has occurred:

P(B/A) = P(A)

P(A and B)

4/8

1/4 1/2 = =

Example:

Given a family with two children, find the probability that both are boys, given that at least one is a boy.

P({gb, bg, bb}) =


P(A and B) =

A = at least one boy


P(A) =

3/4

P({gb, bg, bb} {bb}) = P({bb}) =

= 1/3

1/4

Conditional Probability P(B/A) =

3/4

1/4

B = both are boys

S= {gg, gb, bg, bb}

A = {gb, bg, bb}

B = {bb}

)(

)(

AP

BandAP

)(

)(

AP

BandAP=

Two events are Independent if the occurrence of one of them has no effect on the probability of the other.

P(B/A) = P(B)

Independent Events

or


P(A/B) = P(A)

Example:

A single card is randomly selected from a standard 52-card deck. Given the defined events A and B,

A: the selected card is an ace, B: the selected card is red,find the following probabilities.

a) P(B) =

Independent Events

b) P(A and B) =

a) P(B) b) P(A and B) c) P(B/A)


= 1/2

P({Ah, Ad, Ac, As} {all red}) = P({Ah, Ad}) = 2/52

Events A and B are independent as P(B) = P(B/A).

c) P(B/A) = P(A)

P(A and B)

4/52

2/52 1/2 = =

52

26

If A and B are any two events then

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

Multiplication Rule of Probability - Events Involving “And”


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

If A and B are independent events then

A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles. What is the probability that a red marble is selected and then a blue one without replacement?

P(Red and Blue) = P(Red) P(Blue/Red)

= 4/9 3/8

= 12/72

= 1/8

= 0.1667

Example:

Multiplication Rule of Probability - Events Involving “And”


A jar contains 4 red marbles, 3 blue marbles, and 2 yellow marbles. What is the probability that a red marble is selected and then a blue one with replacement?

P(Red and Blue) = P(Red) P(Blue)

= 4/9 3/9

= 12/81

= 4/27

= 0.148

Example:

11.5 – Expected ValueThe Expected Value of x is the sum of the products of the values of x and their corresponding probabilities.

E(x) = x1 P(x1) + x2 P(x2) + x3 P(x3) + … + xn P(xn)

The expected value is a calculation that serves as the best prediction of a value. It is the probability-weighted average of all possible outcomes.

The expected value of a possible future event assists in making mathematically sound decisions. It is often used when making investments, determining a price for numerous services, prioritizing events, and in calculating Return on Investment.

11.5 – Expected ValueExample:

A third grade class was surveyed regarding the number of hours that they played electronic games each day. The probability distribution is given in the table below:

# of Hours (x) Probability P(x)

0 0.3

1 0.4

2 0.2

3 0.1

Calculate the Expected Value of the quantity of time that a third grader spends each day playing electronic games.

11.5 – Expected ValueE(x) = x1 P(x1) + x2 P(x2) + x3 P(x3) + … + xn P(xn)

# of Hours (x) Probability P(x)

0 0.3

1 0.4

2 0.2

3 0.1

Expected value, E(x) = 0 (0.3) + 1 (0.4) + 2 (0.2) + 3 (0.1)

Expected value, E(x) = 0 + 0.4 + 0.4 + 0.3

Expected value, E(x) = 1.1 hours


Find the expected number of boys for a three-child family. Assume girls and boys are equally likely.

# of Boys Probability Product

x P(x) x P(x)

0 1/8

1 3/8

2 3/8

3 1/8

Expected value, E(x) = 0 + 3/8 + 6/8 + 3/8

Expected value, E(x) = 12/8

Expected value, E(x) = 3/2

Expected value, E(x) = 1.5 boys

0

3/8

6/8

3/8

bbb

bbg

bgb

bgg

gbb

gbg

ggb

ggg


Finding Expected Winnings

Win $7 by rolling a 6 on a single die,Win $1 by rolling any other number.

What are the expected net winnings for the game?

Number Payoff Net P(x) x P(x)

1, 2, 3, 4, 5

6

A player pays $3 to play the following game:


A player pays $3 to play the following game:

Expected value, E(x) = – $10/6 + $4/6

Expected value, E(x) = – $6/6

Expected value, E(x) = – $1


What are the expected net winnings for the game?Number Payoff Net P(x) x P(x)

1, 2, 3, 4, 5 $1 - $2 5/6 - $10/6

6 $7 $4 1/6 $4/6

Finding Expected Winnings

11.5 – Expected Value

Previous Example:

Finding Expected WinningsA player pays $3 to play the following game:

A fair game is one in which the net winnings are zero.


Is this a fair game?

An unfair game against the player has negative expected winnings.

An unfair game in favor of the player has positive expected winnings.

It is an unfair game against the player as it has negative expected winnings (–$1) .

NO


Expected Investment Profits

E(x) = -400 (0.2) + 800 (0.5) + 1300 (.3)

Mark intends to invest $6,000 in one of two companies. His research is presented in the tables below:

What are the expected profits (or loses) for each company?

Company ABC

Profit/Loss (x) Probability P(x)

- $400 0.2

$800 0.5

$1300 0.3

Company PDQ

Profit/Loss (x) Probability P(x)

$600 0.8

$1000 0.2

E(x) = -80 + 400 + 390

E(x) = $710

E(x) = 600 (0.8) + 1000 (0.2)

E(x) = 480 + 200

E(x) = $680Profit Profit

12.1 – Visual Displays of Data

In statistics:

A population includes all of the items of interest.

A sample includes some of the items in the population.

The study of statistics is usually divided into two main areas.

Descriptive statistics: has to do with collecting, organizing, summarizing, and presenting data (information).

Inferential statistics: has to do with drawing inferences or conclusions about populations based on information from samples.

Information that has been collected but not yet organized or processed is called raw data.


Raw data are often quantitative (or numerical), but can also be qualitative (or non-numerical).

Quantitative data: The number of siblings in ten different families: 3, 1, 2, 1, 5, 4, 3, 3, 8, 2

Quantitative data can be sorted in mathematical order. The number siblings can appear as 1, 1, 2, 2, 3, 3, 3, 4, 5, 8

Qualitative data: The makes of five different automobiles: Toyota, Ford, Nissan, Chevrolet, Honda

Frequency Distributions

Frequency Distribution is a method to organize data that includes many repeated items.


It lists the distinct values (x) along with their frequencies (f ).

The relative frequency of each distinct item is the fraction, or percentage, of the data set represented by each item.

Example:

Ten students in a math class were polled as to the number of siblings in their families {3, 2, 2, 1, 3, 4, 3, 3, 4, 2}. Construct a frequency distribution and a relative frequency distribution for the data.


Number x Frequency f Relative Frequency f /n

1

2

3

4

1

3

4

2

1/10 = 0.1

2/10 = 0.2

4/10 = 0.4

3/10 = 0.3

Sum = 10 Sum of f/n = 1

Grouped Frequency Distributions

A Grouped Frequency Distribution is used when data sets contain a large numbers of items.


All data items are assigned to their appropriate classes, and displayed in a table.

The data are arranged into groups, or classes.

1. Make sure each data item will fit into one and only one, class.

2. Make all the classes the same width.3. Make sure that the classes do not overlap.4. Use from 5 to 12 classes.

Guidelines for the Classes of a Grouped Frequency Distribution

Example:Twenty students, selected randomly were asked to estimate the number of hours that they had spent studying in the past week (in and out of class). The responses are below.

Tabulate a grouped frequency distribution and a relative frequency distribution.


15 58 37 42 20 27 36 57 2942 51 28 46 29 58 55 43 40

56 36

The data contains values in the tens, twenties, thirties, forties and fifties.

Five classes: 10-19, 20-29, 30-39, 40-49, 50-59

Hours Frequency f Relative Frequency f /n

12.1 – Visual Displays of Data15 58 37 42 20 27 36 57 2942 51 28 46 29 58 55 43 40

56 36

10 – 19

20 – 29

30 – 39

40 – 49

50 – 59

1

5

3

5

6

1/20 = 0.05 = 5%5/20 = 0.25 = 25%

3/20 = 0.15 = 15%

5/20 = 0.25 = 25%

6/20 = 0.30 = 30%

Sum = 20 Sum of f/n = 1 or 100%

Histogram

The data from the frequency distribution or a grouped frequency distribution can be graphically display using a histogram.

0

1

2

3

4

5

1 2 3 4

Siblings

Fre

quen

cy


A series of rectangles, whose lengths represent the frequencies, are placed next to each other.

Number x

Frequency f

Relative Frequency f /n

1

2

3

4

1

3

4

2

1/10 = 0.1

2/10 = 0.2

4/10 = 0.43/10 = 0.3

Sum = 10 Sum of f/n = 1

Example: Histogram of a Grouped Frequency Distribution

Fre

quen

cy

Hours

0

1

2

3

4

5

6

7

10-19 20-29 30-39 40-49 50-59


Hours Frequency f

10 – 19

20 – 29

30 – 39

40 – 49

50 – 59

1

5

3

5

6

In the table, the numbers 10, 20, 30, 40, and 50 are called the lower class limits.

The numbers 19, 29, 39, 49, and 59 are called the upper class limits.

The class width for the distribution is the difference of any two successive lower (or upper) class limits. The class width is 10.

Frequency Polygon

Data can also be displayed by a frequency polygon.

0

1

2

3

4

5

1 2 3 4

Fre

quen

cy

Siblings


Plot a single point at the appropriate height for each frequency, connect the points with a series of connected line segments and complete the polygon with segments that trail down to the axis.

Line Graph

A line graph is used when it is important to show how data changes with respect to another variable, such as time.


Example: Line Graph

The line graph below shows the stock price of company PCWP over a 6-month span.

Pri

ce in

dol

lars

0

1

2

3

4

5

6

7

8

9

Jan Feb Mar Apr May June

Month

Line Graph


Stem-and-Leaf Displays

The digits to the left of the vertical line (blue region), are the “stems,”


The stem and leaf display is another method to present data.

It preserves the exact data values.

15 58 37 42 20 27 36 57 2942 51 28 46 29 58 55 43 40

56 36

1 5

2 0 7 8 9 9

3 6 6 7

4 0 2 2 3 6

5 1 5 6 7 8 8

The corresponding ones digits (green region) are the “leaves.”

Bar Graphs

A frequency distribution of non-numerical data can be presented in the form of a bar graph.


It is similar to a histogram except that the rectangles (bars) usually are not touching each other and sometimes are arranged horizontally rather than vertically.

Example: Bar Graph

0

1

2

3

4

5

6

7

8

A bar graph is given for the occurrence of vowels in this sentence.

Fre

quen

cy

A E I O U

Vowel

Circle Graphs or Pie ChartA graphical alternative to the bar graph is the circle graph, or pie chart. Each sector or wedge, shows the relative magnitude of the categories. The entire circle measures 360°. The measure of each sector angle should correspond to the percentage of the data being represented.


Example: Expenses

A general estimate of Amy’s monthly expenses are illustrated in the circle graph below.

Other 35%Rent25%

Food 30%

Clothing 10%

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