Download and Probability

Section 6.5 ~
Combining Probabilities
Introduction to Probability and Statistics
Ms. Young ~ room 113
Sec. 6.5
Objective

After this section you will be able to distinguish
between independent and dependent events and
between overlapping and non-overlapping events, and
be able to calculate and and either/or probabilities.
Sec. 6.5
And Probabilities

The probability of two or more events
happening at the same time is known as
an and Probability (or joint probability)
 Ex.
~ Suppose you toss two dice as a single toss.
What is the probability of rolling two 4’s?

You can think of this as rolling one die twice since the
outcomes don’t affect one another
1 1 1
P(double 4's) = P(4)  P(4)   
6 6 36
Sec. 6.5
And Probabilities for Independent Events

An independent event is an event that is not affected by the
probabilities of other events

Common independent variables when finding probabilities:




Rolling dice
Tossing coins
Choosing any item out of a certain number and then replacing that item
prior to the next pick
In general, an And Probability for Independent Events is found
by the following formula:
P( A and B) = P( A)  P( B)

This can be extended to more than two events as long as they
are independent (meaning they do not affect each other)
P( A and B and C ) = P( A)  P ( B )  P (C )...
Sec. 6.5
Example 1

Suppose you toss three fair coins. What is the
probability of getting three tails?

Since the coins are independent, you can multiply the
probabilities of each individual event
1 1 1 1
P(3 tails) = P(tail)  P(tail)  P(tail)    
2 2 2 8
coin 1

coin 2
coin 3
The probability that three tossed coins will all land on tails is
1/8.
Sec. 6.5
And Probabilities for Independent Events

Example 2

Find the probability of drawing three queens in a row if after each
draw you replace the card.

Since the draws are independent because you put the card back, you can
multiply the probabilities of each individual event
P(3 queen's) = P(1 queen)  P(1 queen)  P(1 queen) 


4 4 4
1 1 1
      .000455
52 52 52 13 13 13
The probability that you will draw 3 queen’s in a row is very small, but still
possible
Example 3

Suppose you have a fair coin and a spinner with 5 equal sectors,
labeled 1 thru 5. What is the probability of spinning an even number
AND getting heads?



The probability of getting a heads is 1/2
The probability of the spinner landing on an even number is 2/5
The probability of getting a heads AND landing on an even number is:
1 2 2 1
P(heads & even number)   

2 5 10 5
Sec. 6.5
And Probabilities for Dependent Events

A dependent event is an event that is affected by the
probabilities of the other events


Dependent events typically occur when you choose something at
random and then do not replace it
In general, an And Probability for Dependent Events is
represented by the following formula:
P( A and B) = P( A)  P( B given A)

The “given A” means that you need to take the event A into
consideration


Ex. ~ The probability of you choosing an ace of spades out of a full deck
of cards is 1/52, but if you do not replace that card the probability of
choosing the next card will be out of 51, and so on
This principle can be extended to more than two events:
P( A and B and C ) = P( A)  P( B given A)  P(C given A and B)...
Sec. 6.5
And Probabilities for Dependent Events

Example 4

A batch of 15 memory cards contains 5 defective cards. What is the
probability of getting a defective card on both the first and the
second selection (assuming that the memory cards are not replaced)?
P(2 defectives) = P(defective)  P(defective) 

5 4 1 2 2
   
 .0952
15 14 3 7 21
Example 5

The game of BINGO involves drawing labeled buttons from a bin at
random, without replacement. There are 75 buttons, 15 for each of
the letters B, I, N, G, and O. What is the probability of drawing two
B buttons in the first two selections?
15 14 210
7
P(2 B's) = P( B)  P( B)  


 .0378
75 74 5550 185
Sec. 6.5
And Probabilities for Dependent Events

Example 6

A polling organization has a list of 1,000 people for a telephone survey. The
pollsters know that 433 people out of the 1,000 are members of the
Democratic Party. Assuming that a person cannot be called more than once,
what is the probability that the first two people called will be members of the
Democratic Party?
P(First 2 democratic) = P(democrat)  P(democrat) 

Now suppose we treated those events as being independent. What would the
probability be then?
P(First 2 democratic) = P(democrat)  P(democrat) 


433 432

 .1872
1000 999
433 433

 .1875
1000 1000
Notice that the probabilities are nearly identical
In general, if relatively few items or people are selected from a large
pool, the dependent events can be treated as independent events with
very little error

A common guideline to use for this method is if the sample size is less than
5% of the population size
Sec. 6.5
And Probabilities for Dependent Events

Example 7

A nine person jury is selected at random from a very large
pool of people that has equal numbers of men and women.
What is the probability of selecting an all male jury?

Since we are selecting a small number of jurors from a large pool,
we can treat them as independent events, so
9
1 1 1 1 1 1 1 1 1 1
P(all males) =             .00195
2 2 2 2 2 2 2 2 2 2

The probability of an all male jury is approximately .00195, or
roughly 2/2000
Sec. 6.5
Either/Or Probabilities for Non-Overlapping Events

Two events are non-overlapping (or
mutually exclusive) if they cannot occur
at the same time
 Ex.
~ Suppose you roll a die once and want to find
the probability of rolling a 1 or a 2.



These are considered non-overlapping because you can only
roll a 1 or a 2, not both
The theoretical probability of rolling a 1 or a 2 is 2/6 or
1/3
This can also be found by adding the two probabilities:
1 1 2 1
P(rolling a 1 or a 2) = P(rolling a 1) + P(rolling a 2) =   
6 6 6 3
Sec. 6.5
Either/Or Probabilities for Non-Overlapping Events

In general, an Either/Or Probability for NonOverlapping Events is found by the following
formula:
P( A or B) = P( A)  P( B)

This can be extended to more than two events
as long as they are non-overlapping
P( A or B or C) = P( A)  P( B)  P(C )...
Sec. 6.5
Either/Or Probabilities for Non-Overlapping Events

Example 8
 Suppose
you roll a single die. What is the
probability of getting an even number?

The even outcomes are 2, 4, or 6 and are non-overlapping,
so the probability of rolling an even number can be found
by adding each of the individual events:
1 1 1 3 1
P(2, 4, or 6) = P(2)  P(4)  P(6)     
6 6 6 6 2
Sec. 6.5
Either/Or Probabilities for Overlapping Events

Two events are considered to be overlapping if
they can occur at the same time
 Ex.

~ Suppose you’re interested in knowing the
probability of choosing a dog at random that is
either black or a lab. Since a dog can be a black lab,
that outcome would be considered overlapping since
both events can occur at the same time
Either/Or Probabilities for Overlapping Events are
found using the following formula:
P(A or B ) = P(A) + P(B)  P( A and B)

The reason that you have to subtract P(A and B), which is the
probability that the events will occur together (such as a
black lab), is so that you don’t count the “common” outcome
twice when adding the probabilities
Sec. 6.5
Either/Or Probabilities for Overlapping Events

Example 9

To improve tourism between France and the U.S., the two governments form
a committee consisting of 20 people: 2 American men, 4 French men, 6
American Women, and 8 French women. If you meet one of these people at
random, what is the probability that the person will be either a woman or a
French person?
P(woman or French) 
14
20
probability
of a woman

12
20
probability of
a French person

8
20

probability of
a French woman
18 9

20 10
Sec. 6.5
Either/Or Probabilities for Overlapping Events

Example 10:

Pine Creek is an “average” American town: Of its 2,350 citizens, 1,950 are
white, of whom 11%, or 215 people live below the poverty level. Of the 400
minority citizens, 28%, or 112 people, live below the poverty level. If you
visit Pine Creek, what is the probability of meeting a person who is either a
minority or living below poverty level?
In Poverty
Above Poverty
White
215
1,735
Minority
112
288
400
327
112
615
P(minority or below poverty) 



 .262
2350 2350 2350 2350

The probability of meeting a citizen in Pine Creek that is either a minority or
a person living below poverty level is about 26.2% or about 1 in 4
Sec. 6.5
Either/Or Probabilities

Summary of Probability Formulas: