8-8
8-8 Analyzing
AnalyzingDecisions
Decisions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
8-8 Analyzing Decisions
Warm Up
Find each probability.
1. rolling 2 and tossing heads when rolling a
1
number cube and tossing a coin
12
2. rolling an even number or rolling 5 when
2
rolling a number cube
3
3. not choosing a multiple of 11 when randomly
choosing a whole number from 0 to 99 89
100
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Objectives
Explain that probability can be used to
help determine if good decisions are
made. Use probabilities to analyze
decisions and strategies.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Vocabulary
expected value
Holt McDougal Algebra 2
8-8 Analyzing Decisions
In experiments with numerical outcomes, the
expected value (EV) is the weighted
average of the numerical outcomes of a
probability experiment.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Example 1: Finding Expected Value
The sides of a six-sided number cube are
labeled 1, 1, 3, 3, 9, and 9.
A. What is the expected value of the number cube?
Value of
Side
1
1
3
3
9
9
Probability
1
6
1
6
1
6
1
6
1
6
1
6
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Example 1: Continued
E(V) = 1
1
1 + 1
1 + 1
1
+1
3
+3
9
+9
6
6
6
6
6
6
E(V) = 1 +1+ 3 +3 + 9 + 9
26
=
= 41
3
6
6
B. What is the expected value of rolling two number
cubes, one labeled as described in part A and the
other labeled 1– 6?
2
5
1
4 3
+3 2
=7 6 =7.83
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Check It Out! Example 1
What is the expected value of rolling the six
sided number cube as shown in the net below?
Value of
Side
1
2
2
3
3
5
Probability
1
6
1
6
1
6
1
6
1
6
1
6
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Check It Out! Example 1 continued
E(V) = 1 1 + 2 1 + 2 1 + 3 1 + 3 1 + 5 1
6
6
6
6
6
6
E(V) =
1 +2+ 2+ 3 + 3 + 5
16
2
=
=
2
6
3
6
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Example 2 : Using Expected Value in Real-World
Situations
On a mountain, it takes Sam 2 hours to climb
the southern route, unless there is ice, which
increases the time to 4 hours. It takes him 2.5
hours to climb the eastern route, unless there is
ice, which increases the time to 3 hours. If the
chance of ice is 20% on the southern route and
40% on the eastern route, which route should
Sam take if he wants to finish the climb as soon
as possible?
EV(south) = 0.8(2) + 0.2(4) = 2.4
EV(east) = 0.6(2.5) + 0.4(3) = 2.7
He should take the southern route.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Check It Out! Example 2
Jack can take one of three routes to work each
day. Route A takes 16 minutes, Route B takes 10
minutes, and Route C takes 20 minutes. There is
a 40% chance he will encounter an accident in
Route A, which increases travel time to 25
minutes. There is also a 20% chance he will
encounter a traffic jam if he takes Route B,
which increases his travel time to 40 minutes. He
has a 10% chance of experiencing a delay in
Route C, which increases his travel time to 32
minutes. Which route should Jack take to work
each day?
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Check It Out! Example 2 continued
Route A: 0.60(16) + 0.40(25) = 9.6 + 10 = 19.6
minutes;
Route B: 0.20(40) + 0.80(10) = 8 + 8 = 16 minutes;
Route C: 0.90(20) + 0.1(32) = 21.2 minutes.
He should take Route B.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Example 3: The Monty Hall Problem
In a TV game show, a car key is hidden in one of
five bags. The other bags contain fake keys. Once
the contestant picks a bag, the host, knowing
where the key is located, opens a bag with a fake
key. As the contestant answers questions
correctly, he continues to open bags with fake
keys until two bags remain: one with the car key
and one with a fake key. At this time, he offers
the contestant a chance to switch bags. Find the
expected value of sticking with the original bag
and the expected value of switching bags.
1
EV(sticking) =
5
EV(switching) = 4
5
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Check It Out! Example 3
Mikayla is applying to 3 colleges. She makes
estimates of her chances of being accepted, and
estimates of her chances of receiving financial
aid from each, presented below:
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Check It Out! Example 3 Continued
At which college is she most likely to be both
accepted and receive financial aid?
College A: 0.75 · 0.30 = 0.225
College B: 0.65 · 0.40 = 0.260
College C: 0.70 · 0.45 = 0.315
She has a higher probability of being accepted in
College C with a financial aid.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Lesson Quiz: Part I
Find the expected value for number cubes with
the given sides.
1. 3, 5, 5, 5, 10, 20
8
2. 1, 5, 5, 5, 5, 6
4.5
3. A secretary can use either the copier in her office
or the copier in the hall to make copies of a monthly
newsletter. It takes 75 minutes on the copier in her
office, unless there is a jam, in which case it takes
110 minutes. It takes 60 minutes on the hall copier,
unless it jams, in which case it takes 90 minutes.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Lesson Quiz: Part II
The chance of a jam is 15% for the copier in her
office and 40% for the copier in the hall. Which copier
should she use?
EV(office) = 0.85(75) + 0.15(110) = 80.25
EV(hall) = 0.6(60) + 0.4(90) = 72;
she should use the hall copier.
4. Benjamin applied for three jobs. He has a 40%
chance of being hired at the sandwich shop, a 15%
chance of being hired as a mechanic, and a 60%
chance of being hired as a driver.
Holt McDougal Algebra 2
8-8 Analyzing Decisions
Lesson Quiz: Part III
Also, his chances of being hired as a full-time
employee are 25% at the sandwich shop, 80% as a
mechanic, and 30% as a driver. Which job is he most
likely to be hired and be a full-time employee?
sandwich shop: 0.4(0.25) = 0.1; mechanic: 0.15(0.8)
= 0.12; driver: 0.6(0.3) = 0.18; He is most likely to
get both as a driver.
Holt McDougal Algebra 2