Math 1313 Section 7.5 1 Section

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```					Math 1313     Section 7.3
Example 6: An experiment consists of selecting a card at random from a well-shuffled deck of
52 playing cards. Find the probability that

a. a spade or a queen is drawn.

b. a 7 or an 8 is drawn.

c. a red card or a face card?

Example 7: You are a chief for an electric utility company. The employees in your section cut
down trees, climb poles, and splice wire. You report that of the 128 employees in your department
15 cannot do any of the three (management trainees), 22 can cut trees and climb poles only, 31
can cut trees and splice wire, but not climb poles, 18 can do all three, 4 can cut trees only, 5 can
splice wire, but cannot cut trees or climb poles, and 14 can do exactly one of the three. One of
these employees is selected at random. What is the probability that that employee can do at least 2
of the three jobs mentioned here?

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Math 1313        Section 7.4
Section 7.4: Use of Counting Techniques in Probability

Some of the problems we will work will have very large sample spaces or involve multiple events.
In these cases, we will need to use the counting techniques from the chapter 6 to help solve the
probability problems. In particular, we’ll work with the multiplication principle and combinations.
Let S be a sample space and let E be any event. Then       =

Example 1: Consider the experiment of tossing a fair coin (you may assume it is a uniform sample
space) five times.

a. Find the probability that the coin lands heads exactly 2 times?

b. Find the probability that the coin lands heads either 3 or 5 times.

Popper

An unbiased coin is tossed 16 times. Find the probability of getting exactly 7 times.

a.   0.1746
b.   0.1833
c.   0.2968
d.   0.4375

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Math 1313     Section 7.4
Example 2: From a group of 8 freshmen, 7 sophomores 5 juniors and 3 seniors, what is the
probability that a staff of 2 freshman, 2 sophomores, 2 juniors and 2 seniors will be selected for
the yearbook staff. Assume that each student is equally likely to be chosen.

Example 3: Two cards are selected at random without replacement from a well shuffled deck of
52 playing cards.

a. What is the probability that the two cards drawn are jacks?

b. What is the probability that at least one of the cards a spade?

c. Find the probability that the two cards drawn are red.

d. What is the probability that at least one of the cards is black?

Note that the phrase ”at least one” is a clue that you can use        =   −

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Math 1313     Section 7.4
Example 4: A 6 member is to be formed at random from 9 Democrats and 6 Republicans. Find the
probability that the committee will consist of

a. At least 2 Republicans?

b. Two Democrats and three Republicans?

c. At least one Democrat?

Example 5: An urn contains 25 marbles of which 11 are red and 14 are blue. What is the
probability that a person choosing 4 marbles at random will choose

a. At least 2 red marbles?

b. At most 1 red marble?

c. Exactly 3 blue and one red?

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Math 1313      Section 7.4
Example 6: A department store is shipped one hundred computer controlled toys of which 7 are
defective. A customer selects 8 of these toys, what is the probability that:

a. At least one will be defective?

b. More than 4 will be defective?

c. At most 2 will be defective?

d. Exactly 2 are defective?

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Math 1313       Section 7.4
Example 7: A class contains 30 students, 18 girls and 12 boys. A group of 5 students is chosen at
random from the class to make a presentation to the school board. What is the probability that the
group making the presentation is made up of

a. All boys?

b. More girls than boys?

c. At least 3 girls?

d. At least 1 boy?

e. 3 boys and 2 girls?

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Math 1313     Section 7.5
Section 7.5: Conditional Probability

Example 1: Two cards are drawn without replacement in succession from a well-shuffled deck of
52 playing cards. What is the probability that the second card drawn is an ace, given that the first
card drawn was an ace?

The previous example is an example of conditional probability.
Conditional Probability of an Event
If A and B are events in an experiment and       ≠ 0, then the conditional probability that the
event B will occur given that the event A has already occurred is

∩
|     =

Example 2: Given         = 0.36,         = 0.50, and         ∩   = 0.22. Find    |   .

Example 3: Given        ∩      = 0.39,           = 0.43 and       = 0.59. Find       |   .

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Math 1313        Section 7.5
Example 4: In a certain community a survey showed that 65% of all convenience store shoppers
is the probability that the shopper will buy milk?

Example 5: A pair of fair dice is cast. What is the probability that the sum of the numbers falling
uppermost is 7, if it is known that exactly one of the numbers is a 2?

Popper
Let A and B be events in a sample space S such that       = 0.56,       = 0.45 and
∩    = 0.15. Find      | ).

a.   0.2520
b.   0.2679
c.   0.3333
d.   0.8036

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Math 1313     Section 7.5
Example 6: A pair of fair dice is cast. What is the probability that at least one of the numbers
falling uppermost is a 5, if it is known that the two numbers are different?

The Product Rule
Suppose we know the conditional probability and we are interested in finding.         ∩

Then if      ≠ 0 then       ∩     =           |

In Chapter 6 we used tree diagrams to help us list all outcomes of an experiment. In this Section
tree diagrams will provide a systematic way to analyze probability experiments that have two or
more trials. For example, say we choose one card at random from a well-shuffled deck of 52
playing cards and then go back in for another card. The first trial would be the first draw. The
second trial would be the second draw.

Example 7: Urn 1 contains 4 white and 8 blue marbles. Urn 2 contains 6 white and 9 blue marbles.
One of the two urns is chosen at random with one as likely to be chosen as the other. An urn is
selected at random and then a marble is drawn from the chosen urn. What is the probability that
Urn 1 was chosen and that the chosen marble was blue?

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Math 1313        Section 7.5
Example 8: An urn contains 7 blue and 4 green marbles. Two marbles are drawn in succession
without replacement from the urn. What is the probability that the second marble was green?

Popper
An urn contains 7 green balls and 13 blue balls. Two balls are then drawn in succession. What is
the probability that both balls drawn have a different color if the first ball is replaced before the
second is drawn?

a.   0.4225
b.   0.4550
c.   0.5450
d.   0.2275

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Math 1313     Section 7.5
Example 9: A new lie-detector test has been devised and must be tested before it is put into use.
One hundred people are selected at random and each person draws and keeps a card from a box of
100 cards. Half the cards instruct the person to lie and the other half instruct the person to tell the
truth. The test indicates lying in 85% of those who lied and in 4% of those who did not lie. What is
the probability that for a randomly chosen person the person was instructed not to lie and the test
did not indicate lying?

Example 10: Beauty Girl, a cosmetics company, estimates that 42% of the country has seen its
commercial and if a person sees its commercial, there is a 10% chance that the person will not buy
its product. The company also claims that if a person does not see its commercial, there is a 30%
chance that the person will buy its product. What is the probability that a person chosen at
random in the country will not buy the product?

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Math 1313     Section 7.5
Independent Events
Two events A and B are independent if the outcome of one does not affect the outcome of the
other.

Test for the Independence of Two Events

Two events, A and B, are independent if and only if    ∩       =   ∙

(This formula can be extended to a finite number of events.)

Independent and mutually exclusive does not mean the same thing.

For example, if      ≠ 0 and        ≠ 0 and A and B are mutually exclusive then
∗       ≠ 0
∩    = 0
so            ≠      ∩

Example 11: Determine whether the events A and B are independent. Let         = 0.60,
= 0.35 and      ∩    = 0.21. Then find     ∪      .

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