Download 13-3 Probability and Odds

Advanced Mathematical Concepts
Chapter 13
Lesson 13-3
Example 1
ENTERTAINMENT Each year, about 50,000 people audition to become contestants on a show that
seeks to find the best singer and award him or her a recording contract. Only 12 are chosen to be
finalists on the show. What is the probability of being selected as a finalist?
Use the probability formula. Since 12 people are selected as finalists, s = 12. The denominator,
s + f, represents the total number of people auditioning, those selected as finalists, s, and those not selected, f.
So, s + f = 50,000.
P(12) =
3
12 or
12,500
50,000
s
P(s) = s + f
The probability of any one person being selected as a finalist is
3
or 0.024%.
12,500
Example 2
A cookie jar contains 9 chocolate chip, 5 oatmeal, and 6 sugar cookies.
a. What is the probability that a cookie selected at random will be a sugar cookie?
b. What is the probability that a cookie selected at random will not be chocolate chip?
a. The probability of selecting a sugar cookie is P(sugar). There are 6 ways to select a sugar cookie from the
jar, and 9 + 5 or 14 ways not to select a sugar cookie. So, s = 6 and f = 14.
6
3
P(sugar) = 6 + 14 or 10
s
P(s) = s + f
3
The probability of selecting a sugar cookie is 10.
b. There are 9 ways to choose a chocolate chip cookie. So there are 11 ways not to select a chocolate chip
cookie.
11
11
P(not chocolate chip) = 9 + 11 or 20
11
The probability of not selecting a chocolate chip cookie is 20.
Advanced Mathematical Concepts
Chapter 13
Example 3
A class of 15 children is made up 6 girls and 9 boys. If all their names are placed in a hat and then 4
names are selected from the hat at random, what is the probability that the names chosen will be girls?
There are C(6, 4) ways to choose 4 out of 6 girls, and C(15, 4) ways to select 4 out of 15 children.
C(6, 4)
P(4 girls) = C(15, 4)
6!
2! 4!
1
= 15! or 91
11! 4!
The probability of selecting four girls is
1
.
91
Example 4
An automobile manufacturer determines that for every 20 cars it produces of a particular model, 2 will
have a defective oil tank. If a car dealership has 20 cars of this model on its lot and it sells 4 of them in
one week, what is the probability that at least 1 one of the cars sold has a defective oil tank?
The complement of selecting at least 1 car with a defective oil tank is selecting no cars with defective oil
tanks. That is, P(at least 1 defective oil tank) = 1 - P(no defective oil tank).
P(at least 1 defective oil tank) = 1 - P(no defective oil tank)
C(18, 4)
=1C(20, 4)
3060
= 1 - 4845
 0.3684210526
The probability of selling at least 1 car with a defective oil tank is about 37%.
Advanced Mathematical Concepts
Chapter 13
Example 5
A game at a town fair has children use a fishing rod with a magnet at the end of the fishing line to catch
plastic fish floating in a tank of water. Underneath each fish is either the letter S or L. If a child catches
a fish with the letter S, she may choose a small prize. If a child catches a fish with the letter L, he may
choose a large prize. There are 40 fish with the letter S and 10 fish with the letter L.
a. What is the probability that a child will catch a fish with the letter L?
b. What are the odds that a child will win a large prize?
10 1
a. The probability that a child will catch a fish with the letter L is 50 or 5.
b. To find the odds that a child will win a large prize, you to need to know the probability of a successful
outcome and of a failing outcome.
Let s represent catching a fish with the letter L and f represent not catching a fish with the letter L.
P(s) =
1
5
P(f) =
4
5
Now find the odds.
1
P(s) 5
1
P(f) = 4 or 4
5
1
1
The odds that a child will win a large prize is 4. The ratio 4 is read “1 to 4.”
Advanced Mathematical Concepts
Chapter 13
Example 6
As a fundraiser, a school is holding a raffle. Out of the general pool of participants, 25 are selected to
enter the grand prize drawing. Four grand prizes are to be given to 4 different individuals. If there are
15 men and 10 women selected for the grand prize drawing, and names are drawn at random, what are
the odds that 1 will be male and 3 will be female?
First, determine the total number of possible groups.
C(15, 1)
C(10, 3)
number of groups of 1 male
number of groups of 3 females
Using the Basic Counting Principle we can find the number of possible groups of 1 male and
3 females.
C(15, 1)  C(10, 3) =
15!
10!

or 1800 possible groups
14! 1! 7! 3!
The total number of groups of 4 recipients out of the 25 who qualified is C(25, 4) or 12,650. So, the number
of groups that do not have 1 male and 3 females is 12,650 - 1800 or 10,850.
Finally, determine the odds.
P(s) = 1800
12,650
P(f) =
10,850
12,650
1800
36
12,650
odds =
or 217
10,850
12,650
36
1
Thus, the odds of selecting a group of 3 females and 1 male are 217 or close to 6.