Question 1151739: In a state lottery a seven-digit integer is selected at random. If a player
bets 1 dollar on a particular number, the payoff (if that number is selected)
is $600 minus the $1 paid for the ticket. Let X equal the payoff to the better.
Find the expected value of X.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Let's say we had 7 slots A,B,C,D,E,F,G
The slots will represent the digits of the 7-digit number.
We have 9 choices to fill for slot A. We can choose anything from {1,2,3,4,5,6,7,8,9}
We have 10 choices to fill for slot B. We can choose anything from {0,1,2,3,4,5,6,7,8,9}
Slots C through G will have the same number of choices as slot B.
Here are the number of choices we have for slots A through G
9, 10, 10, 10, 10, 10, 10
There are 9*10*10*10*10*10*10 = 9,000,000 different seven-digit numbers
The smallest of which is 1,000,000 and the largest is 9,999,999
This figure of 9,000,000 can also be found by subtracting the endpoints (1,000,000 and 9,999,999) and then adding 1
9,999,999 - 1,000,000 + 1 = 8,999,999 + 1 = 9,000,000
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Whichever method you use, there are 9,000,000 different seven-digit numbers
There is only one winning ticket
The probability of winning is
1/(9,000,000) = 0.000000111 approximately
The probability of losing is
1-0.000000111 = 0.999999889 approximately
| Outcome | Net Payoff X | P(X) | X*P(X) |
| Win | 599 | 0.000000111 | 0.000066489 |
| Lose | -1 | 0.999999889 | -0.999999889 |
Adding up the X*P(X) values gets us
0.000066489 + (-0.999999889) = -0.9999334
which rounds to -1.00
On average, the person playing the lottery will lose $1
This is effectively the same as saying "on average, the person playing the lottery will not win the jackpot" since the price to play the game is $1.
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