Download Probability Review

Math 141 Lecture Notes
Chapter 7 Review
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E and F are mutually exclusive if E  F = .
Property 1: P(E)  0 for any E
Property 2: P(S) = 1
Property 3: If E and F are mutually exclusive (that is, only one of them can occur,
or equivalently, E  F = ), then
P(E  F) = P(E) + P(F)
Property 4: Addition Rule: If E and F are any two events of an experiment, then
P(E  F) = P(E) + P(F) – P(E  F)
Property 5: Rule of Complements: If E is an event of an experiment and E C
denotes the complement of E, then
P( E C ) = 1 – P(E)
Let S be a uniform sample space and let E be any event. Then,
Number of favorable outcomes in E n( E )
P( E ) 

Number of possible outcomes in S
n( S )
Number of elements in A  B n( A  B)

Number of elements in A
n( A)
P( A  B)
P( B | A) 
P( A)
P( A  B)  P( A)  P( B | A)
P( E  F  G )  P( E )  P( F | E )  P(G | E  F )
If A and B are independent events, then P( A | B)  P( A)
and
P( B | A)  P( B) .
Two events A and B are independent if and only if P( A  B)  P( A)  P( B) .
P( A)  P( D | A)
P ( A | D) 
P( A)  P( D | A)  P( B)  P( D | B)  P(C )  P( D | C )
Pr oduct of probabilit ies along the lim b through A
P( A | D) 
Sum of products of the probabilit ies along each lim b ter min ating at D
P( B | A) 
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
1. Let E and F be two mutually exclusive events, P(E) = .3 and P(F) = .4. Compute
a. P(EF)
b. P(EF)
c. P( E C )
d. P( E C  F c )
e. P( E C  F c )
2. Let E and F be two events of an experiment with sample space S. Suppose
P(E) = .4, P(F) = .3, and P(EF) = .2. Compute:
a. P(EF)
b. P( E C  F )
c. P( E C  F c )
3. A die is loaded and an experiment of casting the die and observing which number
falls uppermost is given by
Simple Event
1
2
3
4
5
6
Number of Times Event Occurs
14
20
24
17
10
15
a. Write the probability distribution.
b. What is the probability that the number is even?
c. What is the probability that the number is either a 1 or a 5?
d. What is the probability that the number is less than three?
4. An urn contains seven red, four black, and five green balls. If two balls are
selected at random without replacement from the urn, what is the probability that a
red ball and a black ball will be selected?
5. The quality-control department of Stem Communications has determined that 2% of
cartridges sold have video defects,, 1.2% have audio defects, and .8% have both
audio and video defects. What is the probability that a cartridge purchased by a
customer
a. Will have a video or audio defect?
b. Will not have a video or audio defect?
6. Let E and F be two events and supposed P(E) = .45, P(F) = .58, and P(EF) =
.70. Find P(E|F).
The tree diagram below represents an experiment consisting of two trials.
E
.7
A
.3
.4
EC
E
.3
.6
B
.4
.3
EC
E
.5
C
.5
EC
7. Using the tree diagram above, find the given probability.
a. P(AE)
b. P(BE)
c. P(CE)
d. P(A|E)
e. P(E)
8. An experiment consists of tossing a fair coin three times and observing the
outcomes. Let A be the event that at least two heads are thrown and B the event
that at most one tail is thrown.
f. Find P(A)
g. Find P(B)
h. Are A and B independent events?
9. A pair of fair dice is cast. What is the probability that the sum of the numbers
falling uppermost is 8 if it is known that the two numbers are different?
10. Three cards are drawn at random without replacement from a standard deck of 52
playing cards. Find the probability of each of the given events.
a. All three cards are kings.
b. The first and third cards are black.
c. The second card is red, given that the first card was a spade.
d. The second card is a diamond, given that the first card was red.
11. In a manufacturing plant, three machines, A, B, and C, produce 45%, 30%, and
25%, respectively, of the total production. The company’s quality-control
department has determined that 1% of the items produced by machine A, 2% of
the items produced by machine B, and 2.5% of the items produced by machine C
are defective. If an item is selected at random and found to be defective, what is
the probability that it was produced by machine B?
12. The sales department of a drug company released the accompany data
concerning the sales of a certain pain reliever manufactured by the company.
Pain Reliever
Group I (capsule)
Group II (tablet)
Percent of Drug Sold
.53
.47
Percent of Group Sold in
Extra-Strength Dosage
.38
.31
If a customer purchased the extra-strength dosage of this drug, what is the
probability that it was in capsule form?