Data Interpretation & Probability
4.2 Expected Value
Mathematical expectation and expected value are terms used to describe the expected winning from a contest or game. Expected value is calculated to determine prizes for contests and insurance policies. You must know the probabilities of all possible outcomes to calculate the expected value.
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Example 1:
A six sided die is rolled. If the die is rolled and lands on a 1,2,3, the player wins nothing. If the die lands on a 4 or 5, the player wins $2. If the die lands on a 6, the player wins $10. If you play this game what is the expected value of the game?
Solution:
|
EVENT |
P(event) |
win |
|
1, 2 or 3 |
3/6 |
0 |
|
4, or 5 |
2/6 |
2 |
|
6 |
1/6 |
20 |
Expected Value = 
Therefore the expected value of this game is four dollars.
The expected value of $4 means an average of $4 for each game played. So if the game is played 100 times the expected win is $400.
This is just expected value. After playing three games, the expected value is $12, but the player could win as little as nothing or as much as $60.
If $1 is charged to play the game, the expected value is reduced to $3.
|
EVENT |
P(event) |
win |
|
1, 2 or 3 |
3/6 |
-1 |
|
4, or 5 |
2/6 |
1 |
|
6 |
1/6 |
19 |
Expected Value = 
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Example 2:
A State lottery has 51 numbers in which 6 will be the winning numbers. The cost of playing the lottery is $1. A match of 6 out of 6 wins the grand prize of $1,000,000. A match 3 of the winning numbers wins the prize of $5. A match 4 or 5 of the winning numbers wins the prize of $130. Including the cost of the game what is the expected value of this game?
|
Event |
P(event) |
Win |
Win with cost |
|
0 |
|
0 |
-1 |
|
1 |
|
0 |
-1 |
|
2 |
|
0 |
-1 |
|
3 |
|
5 |
4 |
|
4 |
|
130 |
129 |
|
5 |
|
130 |
129 |
|
6 |
|
1,000,000 |
999,999 |
The expected value is:
(.45)(-1)+(.41)(-1)+(.12)(-1)+(.02)(4)+(.0008)(129)+(.00001)(129)+(.00000006)(999,999)
= - .74
Therefore the expected value is a loss of 74 cents.
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Example 3:
A bank teller’s cash drawer contains 100 one-dollar bills, 50 five-dollar bills, 50 ten-dollar bills, 25 twenty-dollar bills. What is the expected value of a bill chosen at random from the drawer.
Try to this one on your own before looking at the answer.
Solution:
|
Value of Bill |
Quantity |
Total Value |
P(event) |
|
1 |
100 |
100 |
100/225 |
|
5 |
50 |
250 |
250/225 |
|
10 |
50 |
500 |
500/225 |
|
20 |
25 |
500 |
500/225 |
The expected value is:
Therefore $6 dollars is the expected value.
In a particular lottery, 161 tickets will be randomly selected from among the 250,000 tickets sold. One ticket will win $10,000, ten tickets will win $1000, 50 tickets will win $200, and 100 tickets will win $100.
A) Set up the table that shows the Probabilities and Wins if each ticket costs $1?
Answer
Answer
Student #1:
|
#of tickets |
P(event) |
Win w/costs |
|
1 |
1/250,000 |
9,999 |
|
10 |
10/250,000 |
999 |
|
50 |
50/250,000 |
199 |
|
100 |
100/250,000 |
99 |
|
249,839 |
. 9994 |
-1 |
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