holt mcdougal algebra 2 analyzing decisions explain that probability can be used to help determine...
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Holt McDougal Algebra 2
Analyzing Decisions
Explain that probability can be used to help determine if good decisions aremade. Use probabilities to analyzedecisions and strategies.
Objectives
Holt McDougal Algebra 2
Analyzing Decisions
expected value
Vocabulary
Holt McDougal Algebra 2
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
Analyzing Decisions
Holt McDougal Algebra 2
Analyzing DecisionsExample 1: Finding Expected Value
The sides of a six-sided number cube are labeled 1, 1, 3, 3, 9, and 9.
Value of Side
Probability
A. What is the expected value of the number cube?
1
16
16
16
16
16
16
1 3 3 9 9
Holt McDougal Algebra 2
Analyzing DecisionsExample 1: Continued
B. What is the expected value of rolling two number cubes, one labeled as described in part A and the other labeled 1– 6?
234 1
2+3 56=7 =7.83
1 16
+ + + + +E(V) = 1 16
3 16
3 16
9 16
9 16
+1 + += =
E(V) = 1 +3 3 9+ 96
266
Holt McDougal Algebra 2
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
Probability
1
16
16
16
16
16
16
2 2 3 3 5
Holt McDougal Algebra 2
Analyzing Decisions
Check It Out! Example 1 continued
E(V) = 2 16
2 16
3 16
3 16
5 16
1 1 +6
+ + + +
E(V) = 1 +2 3 3+ 56
166
+ ++2 = =
Holt McDougal Algebra 2
Analyzing DecisionsExample 2 : Using Expected Value in Real-World Situations
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.
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?
Holt McDougal Algebra 2
Analyzing DecisionsCheck 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
Analyzing DecisionsCheck 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
Analyzing DecisionsExample 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.
EV(sticking) = 15
EV(switching) = 45
Holt McDougal Algebra 2
Analyzing DecisionsCheck 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
Analyzing Decisions
Check It Out! Example 3 Continued
College A: 0.75 · 0.30 = 0.225 College B: 0.65 · 0.40 = 0.260College C: 0.70 · 0.45 = 0.315
She has a higher probability of being accepted in College C with financial aid.
At which college is she most likely to be both accepted and receive financial aid?
Holt McDougal Algebra 2
Analyzing Decisions