Transcript The probability of an odd sum is ____ . 6.5 Find Expected Values

6.5
Find Expected Values
Example 1 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.
Solution
The possible outcomes are 2 + 2, 2 + 3, 2 + 4, 3 + 2, 3 + 3, 3 + 4,
4
4 + 2, 4 + 3, 4 + 4. The probability of an even sum is ____. The
9
5
probability of an odd sum is ____.
9
4
5 7
 1  
Player A : E  __
3   ____
9
9
9 ________
6.5
Find Expected Values
Checkpoint. The outcome values and their
probabilities are given. Find the expected value.
1. Outcome value, x
Probability, p
2
3
1
0.20 0.45 0.35
2 0.20  3 0.45   1 0.35
 1.4
6.5
Find Expected Values
Checkpoint. The outcome values and their
probabilities are given. Find the expected value.
2. Outcome value, x
Probability, p
$8
$4  $2
0.10 0.25 0.65
$8  0.10  $4  0.25   2 0.65
 $0.50
6.5
Find Expected Values
Example 2 Use 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?
Solution
Step 1 Find the probability of each outcome. Because each
question has 1 right answer and 2 wrong answers the
probability of guessing correctly is ____
1/3 and the
probability of guessing incorrectly is ____.
2/3
6.5
Find Expected Values
Example 2 Use 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?
Solution
Step 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.
1
2


 $6   
10    _____
E  $____
 3
3
2
3
Because the expected value is negative, it is not to your advantage to guess.
6.5
Find Expected Values
Example 3 Find expected value
Theater 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
Step 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 value
Theater 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
Step 2 Find the probability of each outcome. There are 1000
tickets sold, and the probability of winning one of the
prizes is 1 . Because there are 2 prizes there are 2
998 losing tickets.
1000 winning tickets and _____
998
So, the probability you will not win a prize is
.
1000
6.5
Find Expected Values
Example 3 Find expected value
Theater 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
Step 3 Summarize the information in the table.
Gain, x
$94
$44  $6
Probability, p
1
1000
1
1000
998
1000
6.5
Find Expected Values
Example 3 Find expected value
Theater 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 $6. The ticket collectors will take the ticket
from the first 1000 guests, and after the movie they will
randomly choose one ticket. If the number chosen matches
the number on your ticket stub, you will win 1st or 2nd prize.
What is the expected value of your gain?
Solution
Step 4 Find the expected value by finding the sum of each
outcome multiplied by its corresponding probability.
 1 
 1 
 998 
E $___
 $6 
94  
  ____
  _____

$44  
 1000 
 1000 
 1000 
$5.85
 ______
6.5
Find Expected Values
Example 3 Find expected value
Theater 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 _________.
 $5.85 This means that
you expect to ______
lose an average of _________
$5.85 for each ticket
you buy.
6.5
Find Expected Values
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.
2 8 8
1
E  $8     $4      0
3 3 3
3
There is no advantage or disadvantage to guessing.
6.5
Find Expected Values
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?
1 
1 
 1 


E $1492 
  $792 
 $292 

 5000 
 5000 
 5000 
 4997 
  $8
   $7.48
 5000 
6.5
Find Expected Values
Pg. 375, 6.5 #1-8