The possible outcomes are 2 + 2, 2 + 3, 2 + 4, 3 + 2, 3 + 3, 3 + 4, 4 + 2, 4 + 3, 4 + 4. The probability of an even sum is ____ . The probability of an odd sum is ____ . 6.5 Find Expected Values Example 1Find an expected value Find an expected value Game Consider a game in which two players each choose an integer from 2 to 4. If the sum of the two integers is odd, then player A scores 3 points and player B loses 1 point. If the sum is even, then player B scores 3 points and player A loses 1 point. Find the expected value for player A. Soluti on ________ 9 5 ____ 9 4 __ E : A Player 9 4 3 9 7 9 5 1
Transcript
The possible outcomes are 2 + 2, 2 + 3, 2 + 4, 3 + 2, 3 + 3, 3 + 4,
4 + 2, 4 + 3, 4 + 4. The probability of an even sum is ____
. The
probability of an odd sum is ____
.
6.5 Find Expected Values
Example 1 Find an expected valueFind an expected value
Game Consider a game in which two players each choose an integer from 2 to 4. If the sum of the two integers is odd, then player A scores 3 points and player B loses 1 point. If the sum is even, then player B scores 3 points and player A loses 1 point. Find the expected value for player A.
Solution
________9
5____
9
4__ E :APlayer
9
4
39
79
5
1
6.5 Find Expected ValuesCheckpoint. The outcome values and their Checkpoint. The outcome values and their probabilities are given. Find the expected value.probabilities are given. Find the expected value.1. Outcome value, x
Probability, p
2 3 120.0 45.0 35.0
20.02 45.03 35.01 4.1
6.5 Find Expected ValuesCheckpoint. The outcome values and their Checkpoint. The outcome values and their probabilities are given. Find the expected value.probabilities are given. Find the expected value.2. Outcome value, x
Probability, p
8$ 4$ 2$10.0 25.0 65.0
10.08$ 25.04$ 65.02 50.0$
6.5 Find Expected Values
Example 2 Use expected valueUse expected value
Game Show You participate in a game show in which you respond to questions that have 3 possible answers. You gain $10 for each correct answer, and lose $6 for each incorrect answer. Every question must be answered. If you do not know the answer to one of the questions, is it to your advantage to guess the answer?SolutionStep 1 Find the probability of each outcome. Because each
question has 1 right answer and 2 wrong answers the
probability of guessing correctly is ____ and the
probability of guessing incorrectly is ____.
1/3
2/3
6.5 Find Expected Values
Example 2 Use expected valueUse expected value
Game Show You participate in a game show in which you respond to questions that have 3 possible answers. You gain $10 for each correct answer, and lose $6 for each incorrect answer. Every question must be answered. If you do not know the answer to one of the questions, is it to your advantage to guess the answer?SolutionStep 2 Find the expected value of guessing an answer. Multiply
the money gained or lost by the corresponding probability, then find the sum of these products.
3
2_____
3
1____ E
10$ 6$ 2
3Because the expected value is negative, it is not to your advantage to guess.
6.5 Find Expected Values
Example 3 Find expected valueFind expected valueTheater A movie theater is giving away a $100 prize and a $50 prize. To enter the drawing, you need to simply buy a movie ticket for $6. The ticket collectors will take the ticket from the first 1000 guests, and after the movie they will randomly choose one ticket. If the number chosen matches the number on your ticket stub, you will win 1st or 2nd prize. What is the expected value of your gain?
SolutionStep 1 Find the gain for each prize by subtracting the cost of the
ticket from the prize money.
6.5 Find Expected Values
Example 3 Find expected valueFind expected valueTheater A movie theater is giving away a $100 prize and a $50 prize. To enter the drawing, you need to simply buy a movie ticket for $6. The ticket collectors will take the ticket from the first 1000 guests, and after the movie they will randomly choose one ticket. If the number chosen matches the number on your ticket stub, you will win 1st or 2nd prize. What is the expected value of your gain?
SolutionStep 2 Find the probability of each outcome. There are 1000
tickets sold, and the probability of winning one of the
prizes is . Because there are 2 prizes there are 2 winning tickets and _____ losing tickets.
1
1000 998
So, the probability you will not win a prize is . 998
1000
6.5 Find Expected Values
Example 3 Find expected valueFind expected valueTheater A movie theater is giving away a $100 prize and a $50 prize. To enter the drawing, you need to simply buy a movie ticket for $6. The ticket collectors will take the ticket from the first 1000 guests, and after the movie they will randomly choose one ticket. If the number chosen matches the number on your ticket stub, you will win 1st or 2nd prize. What is the expected value of your gain?
SolutionStep 3 Summarize the information in the table.
Gain, x
Probability, p1000
1
1000
1
1000
99894$ 44$ 6$
6.5 Find Expected Values
Example 3 Find expected valueFind expected valueTheater A movie theater is giving away a $100 prize and a $50 prize. To enter the drawing, you need to simply buy a movie ticket for $6. The ticket collectors will take the ticket from the first 1000 guests, and after the movie they will randomly choose one ticket. If the number chosen matches the number on your ticket stub, you will win 1st or 2nd prize. What is the expected value of your gain?
SolutionStep 4 Find the expected value by finding the sum of each
outcome multiplied by its corresponding probability.
1000
998_____
1000
1____
1000
1___ E 94$ 44$ 6$
______ 85.5$
6.5 Find Expected Values
Example 3 Find expected valueFind expected valueTheater A movie theater is giving away a $100 prize and a $50 prize. To enter the drawing, you need to simply buy a movie ticket for $6. The ticket collectors will take the ticket from the first 1000 guests, and after the movie they will randomly choose one ticket. If the number chosen matches the number on your ticket stub, you will win 1st or 2nd prize. What is the expected value of your gain?
Solution
The expected value of your gain is _________. This means that
you expect to ______ an average of _________ for each ticket
you buy.
85.5$lose 85.5$
6.5 Find Expected ValuesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 3. In Example 2, suppose you gain $8 for each
correct answer and lose $4 for each incorrect answer. Find the expected value and then determine if it is to your advantage to guess on a particular question.
E 8$ 4$
3
1
3
2
3
8
3
8 0
There is no advantage or disadvantage to guessing.
6.5 Find Expected ValuesCheckpoint. Complete the following exercises. Checkpoint. Complete the following exercises. 4. There is a prize drawing for home
electronics. Tickets are $8. There are a total of 5000 tickets sold for the drawing. The three prizes are a new computer worth $1500, a high-definition TV worth $800, and a stereo system worth $300. If you buy one ticket, what is the expected value of your gain?
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