Section 6.5 ~ Combining Probabilities

Introduction to Probability and StatisticsMs. Young ~ room 113

Objective

Sec. 6.5

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.

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

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1 1 1(double 4's) = (4) (4)

6 6 36P P P

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:

This can be extended to more than two events as long as they are independent (meaning they do not affect each other)

Sec. 6.5

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

( and and ) = ( ) ( ) ( )...P A B C P A P B P C

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

The probability that three tossed coins will all land on tails is 1/8.

1 1 1 1(3 tails) = (tail) (tail) (tail)

2 2 2 8P P P P

coin 1 coin 2 coin 3

Example 1

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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

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:

4 4 4 1 1 1(3 queen's) = (1 queen) (1 queen) (1 queen) .000455

52 52 52 13 13 13P P P P

Sec. 6.5

1 2 2 1(heads & even number)

2 5 10 5P

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:

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:

Sec. 6.5

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

( and and ) = ( ) ( given ) ( given and )...P A B C P A P B A P C A B

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)?

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?

5 4 1 2 2(2 defectives) = (defective) (defective) .0952

15 14 3 7 21P P P

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15 14 210 7(2 B's) = ( ) ( ) .0378

75 74 5550 185P P B P B

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?

Now suppose we treated those events as being independent. What would the probability be then?

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

433 432(First 2 democratic) = (democrat) (democrat) .1872

1000 999P P P

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433 433(First 2 democratic) = (democrat) (democrat) .1875

1000 1000P P P

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

The probability of an all male jury is approximately .00195, or roughly 2/2000

91 1 1 1 1 1 1 1 1 1

(all males) = .001952 2 2 2 2 2 2 2 2 2

P

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:

Sec. 6.5

1 1 2 1(rolling a 1 or a 2) = (rolling a 1) + (rolling a 2) =

6 6 6 3P P P

Either/Or Probabilities for Non-Overlapping Events In general, an Either/Or Probability for

Non-Overlapping Events is found by the following formula:

This can be extended to more than two events as long as they are non-overlapping

Sec. 6.5

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

( or or ) = ( ) ( ) ( )...P A B C P A P B P C

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(2,4,or 6) = (2) (4) (6)

6 6 6 6 2P P P P

Either/Or Probabilities for Non-Overlapping Events

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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:

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

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

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?

Either/Or Probabilities for Overlapping Events

Sec. 6.5

14 12 8 18 9(woman or French)

20 20 20 20 10P

probabilityof a woman

probability of a French person

probability of a French woman

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?

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

Either/Or Probabilities for Overlapping Events

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400 327 112 615(minority or below poverty) .262

2350 2350 2350 2350P

In Poverty Above Poverty

White 215 1,735

Minority 112 288

Summary of Probability Formulas:

Either/Or Probabilities

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