Probability

Pop Quiz 

 

1. Who is George W. Bush's favorite Friend?

BushThe Cast of Friends

a. Chandler

b.  Joey

c.  Monica

d.  Pheobe

e.  Rachael

f.  Ross

2. How many cousins does Professor DeMaio have?

a. 1

b. 2

c. 3

d. 4

e. 5

For both questions random guessing must be employed. The probability of a correct answer in question number 1 is 1/6 or about  16.67%. The probability of a correct answer in question number 2 is 1/5 or 20%. This concept of counting and dividing is 
the heart of computing probabilities.

 

 

The process of making an observation or taking a measurement of one or more experimental units is called an experiment.  Outcome of experiments 
are called events. Events are abbreviated with a capitol letter.

example: An experiment consists of randomly picking a card from a 
standard deck of playing cards (no jokers). 
Some possible outcomes of this experiment are listed below.
 

A - The 8§ is selected.

B - A red card is selected.

C - The J© is not selected.

 
Why do we consider small games of chance when studying probability?
Click here to find out.

 

example: An experiment consists of watching ten different movies made after 1982 
and recording if a Coke product appears in the film. 
List some possible outcomes of this experiment.

 A scene from 'E.T. The Extra-Terrestrial'

 

The probability of an event A, denoted P(A), is a number between 0 and 1 that measures the likelihood that event A will occur when the experiment is performed. 
If P(A) is 0 then A is guaranteed not to occur. 
If P(A) is 1 then A is guaranteed to occur.

 

 

 

Two events A and B are mutually exclusive (or disjoint
if they cannot occur at the same time.

example: Let an experiment consist of rolling a pair of fair dice. 
Consider the following events.

A - The sum of the two dice is an even number.  

B - The sum of the two dice is an odd number.


C
- The sum of the two dice is 7.
  

 

Are events A and B mutually exclusive? Explain.

Are events A and C mutually exclusive? Explain.

Are events B and C mutually exclusive? Explain.

 

Let an experiment consist of a collection of s mutually exclusive and equally likely events. This collection is called the sample space. Furthermore suppose exactly n of the events result in event A. Then the probability that event A will occur is

.

example: An experiment consist of flipping a fair coin twice. Compute the probabilities of the following events.

A - Exactly one head is observed.

B - At least one head is observed.

C - No tails are observed.

The first step is to construct the sample space. In this case the sample space is as follows.

 
HH HT
TH TT

P(A) is 2/4 or 50%, P(B) is 3/4 or 75% and P(C) is 1/4 or 25%. 

 

example: A pair of fair dice are rolled. Compute the probabilities of the following events.

A - The sum of the two dice is 7.

B - The sum of the two dice is 5

C - The sum of the two dice is an even number.

Once again, the first step is to construct the sample space.

 
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)
(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)
(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)
(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)
(1,5) (2,5) (3,5) (4,5) (5,5) (6,5)
(1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

P(A) is 6/36, P(B) is 4/36 and P(C) is 18/36.

 

Homework

Section 3-2: 2-5, 7, 9, 10, 13, 15, 17, 20, 28

 

Addition Rule for  Mutually Exclusive Events

If A and B are mutually exclusive events then P(A or B) = P(A) + P(B).

 

example: The probability that John makes a B in Math 1107 is 35%. 
The probability that John makes a C in Math 1107 is 27%. 
What is the probability that John makes a B or C in Math 1107? 

These events are clearly mutually exclusive and the 
probability that John makes a B or C in Math 1107 is 35% + 27% = 62%.

 

example: A fair coin is flipped twice. 
Let event A be the event that you observe at least 1 head. 
Let event B be the event that you observe at least 1 tail. 
Compute P(A), P(B), P(A and B) and P(A or B).

 

 

General Addition Rule

It is always true that P(A or B) = P(A) + P(B) - P(A and B).

 

 

Complements

The complement of an event A, denoted , is the event that A does not occur.

example: A fair coin is flipped twice. 
Let event A be the event that you observe at least 1 head.

Describe .

Compute P(A) and P( ).

 

example: A pair of fair dice are rolled. Let A be the event that the sum of 10 is rolled.

Describe .

Compute P(A) and P( ).

 

 

Fact: P(A) + P( ) = 1.  This fact quickly implies that 

 
and 
P( ) = 1 - P(A).

Homework

Section 3-3: 1, 3, 4, 7-20, 25, 26

Consider the experiment of randomly picking a card from a standard 
d

A - The suit § is selected.

B - The rank King is selected.

Compute

P(A) and P( )

 

P(B) and P(

 

P(A and B)

 

P(A or B)


P(A and B)


P(A and B)


 

 

 

Conditional Probability and Independence

 

On the first day of spring training you are asked, 
"What is the probability that the Braves win the World Series this year?"

 

Halfway through the year when the Braves have won 
70% of their games and have no injuries, you are asked, 
"What is the probability that the Braves win the World Series this year?"

 

At the end of the regular season, the Braves have clinched a playoff birth, 
have no injuries and you are asked, 
"What is the probability that the Braves win the World Series this year?"

 

The Braves have a 3-0 games lead and are up 15-0 at the 
top of the ninth inning in game 4 of the World Series, have no injuries and 
you are asked, 
"What is the probability that the Braves win the World Series this year?"

 

Why do your answers change? You are repeatedly asked the same question.

The probability of event A, given that event B has occurred, is called the conditional probability of A given B, denoted by P(A|B).

 

example: A pair of dice are thrown one at a time.

Let A be the event that the sum of 9 is rolled.
Let B be the event that the first die thrown is a 2.
Let C be the event that the first die thrown is a 5.
Let D be the event that the sum of 7 is rolled.

 

i. What is the probability the sum of the dice is 9?

ii. What is the probability the sum of the dice is 9, given that the first 
die rolled is 2?


iii. What is the probability the sum of the dice is 9, given that the first 
die rolled is 5?


iv. What is the probability the sum of the dice is 7?


v. What is the probability the sum of the dice is 7, given that the first 
die rolled is 2?


vi. What is the probability the sum of the dice is 7, given that the first 
die rolled is 5?


Compare and contrast the first three probabilities with the last three probabilities.

Can I use conditional probability to make money playing blackjack?
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Two events A and B are said to be independent if 
or .

Note!! Both of these conditions are equivalent. 
You cannot have one without the other.

 

 

Multiplication Rule for Independent Events

If A and B are independent events then P(A and B) = P(A) * P(B).

 

example: The probability that on any given morning 85 south near the connector 
will be slow is 35%.
The probability that on any given 
morning the top end of 285 near Dunwoody will be slow is 57%. 
These events are independent of one another. What is the probability 
that both 85 south near the connector and the top end of 285 
near Dunwoody will be slow tomorrow morning?

P(85 south near the connector and the top end of 285 near Dunwoody will be slow) = P(85 south near the connector will be slow) * 
P
(the top end of 285 near Dunwoody will be slow) = 
.35*.57 = .1995 


Approximately, 15% of all human beings are left handed. 
What is the probability that three randomly selected people are all left-handed?

 

 

 

An box contains 3 white balls, 4 red balls and 5 black balls. 
A ball is picked, its color recorded and returned to the box. Another ball is 
then selected and its color recorded.

Find the probability that 2 black balls are selected.

 


 

 

 

An box contains 3 white balls, 4 red balls and 5 black balls. 
A ball is picked, its color recorded but is not returned to the box. 
Another ball is then selected and its color recorded.

In this case, selections of the balls are not independent events.  

General Multiplication Rule

It is always true that P(A and B) = P(A) * P(B|A).

An box contains 3 white balls, 4 red balls and 5 black balls. 
A ball is picked, its color recorded but is not returned to the box. 
Another ball is then selected and its color recorded.


Find the probability that 2 black balls are selected.


Find the probability that 2 balls of the same color are selected.

 

 

 

If a sample size is no more than 5% of the size of the population, 
treat the selections as being independent 
even if the selections are made without replacement 
and are technically dependent.

 

The defect rate for manufacturing a cd is 2%.  If a particular store 
orders 800 copies of this cd, what is the probability that the first 20
cd's sold are free of defects?

 

 

Homework

Section 3-4: 1-4, 8, 9, 12, 13, 17, 19, 20, 25, 26, 27

 

Complements:  The Probability of "at least one"

The event of "at least one" is equivalent to "one or more."

 

 

Note that if event A is the event of "at least one" then the complement of A is "none."

 

A die is independently rolled 5 times. What is the probability that the number 3 appears at least once?

 

 

According to Nielson Media Research, 30% of all televisions are tuned to 
NFL Monday Night Football
when it is televised. 
Assuming that this show is being broadcast and that the 
televisions are randomly selected, 
find the probability that at least 1 of 15 televisions is tuned to 
NFL Monday Night Football
.

 

Homework

 Section 3-5: 1-4, 7-10, 15, 17, 23