Question 1199817: The student council is hosting a drawing to raise money for scholarships. They are selling tickets for $6 each and will sell 900 tickets. There is one $3,000 grand prize, two $300 second prizes, and fifteen $10 third prizes. You just bought a ticket. Find the expected value for your profit. Round to the nearest cent.
$_______
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Method 1
X = net gain
Define the four events:
A = win the 1st prize $3000
B = win the 2nd prize $300
C = win the 3rd prize $10
D = no winnings
If you paid $6 and win $3000, then the net winnings is 3000-6 = 2994. The chances of this happening is 1 out of 900, aka 1/900.
If you paid $6 and win $300, then the net winnings is 300-6 = 294. The chances of this happening is 2/900.
If you paid $6 and win $10, then the net winnings is 10-6 = 4. The chances of this happening is 15/900.
There are 1+2+15 = 18 winning tickets.
That means there are 900-18 = 882 non-winning tickets.
The chances of not winning any money is 882/900.
Probability Distribution Table:
| Event | X | P(X) | | A | 2994 | 1/900 | | B | 294 | 2/900 | | C | 4 | 15/900 | | D | -6 | 882/900 |
where X = net gain
Form a fourth column X*P(X) to multiply each X value with its corresponding probability P(X)
| Event | X | P(X) | X*P(X) | | A | 2994 | 1/900 | 2994/900 | | B | 294 | 2/900 | 588/900 | | C | 4 | 15/900 | 60/900 | | D | -6 | 882/900 | -5292/900 |
Add up the results in the X*P(X) column to get the expected value.
2994/900 + 588/900 + 60/900 + (-5292/900)
(2994+588+60-5292)/900
-1650/900
-1.83333333333333
-1.83
You should expect to lose approximately $1.83 per ticket on average.
Answer: -1.83
Don't forget about the negative sign to indicate a loss.
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Method 2
Let's say that you purchased all 900 tickets.
1 ticket costs 6 dollars.
900 tickets cost 900*6 = 5400 dollars.
So far you are down $5400.
But the one grand prize $3000, two $300 prizes, and fifteen $10 prizes add up to:
1*3000+2*300+15*10 = 3750 dollars
The net change is
amountIn - amountOut = 3750-5400 = -1650
Meaning you are down $1650
The loss per ticket is: 1650/900 = 1.83 after rounding to two decimal places.
Answer: -1.83
Don't forget about the negative sign to indicate a loss.
Method 2 is much easier to deal with, since we avoid messy fractions.
However, method 1 is a standard approach that many math textbooks and teachers use.
It's good to know how to do both methods.
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