NYS Lottery's Pick 10 game: You pick 10 integers between 1 and 80 (including 1 and 80). The state picks 20 numbers. If all 10 of your numbers are among the state's 20 numbers you win $500,000. (Note that the state won't give $500,000 to each winner if too many people get 10 numbers among the 20 but we will assume that you always get the $500,000 if you match the numbers.) On the webpage you will find the following information:
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Winning Numbers Matched
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Prize
|
Chances of Winning
|
|
10
|
$500,000*
|
1 in: 8,911,711
|
|
9
|
$6,000
|
1 in: 163,381
|
|
8
|
$300
|
1 in: 7,384
|
|
7
|
$40
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1 in: 621
|
|
6
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$10
|
1 in: 87
|
|
0
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$4
|
1 in: 22
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a) Show how you get the probabilities of winning all of the above amounts.
b) Find the expected winnings.
c) Find the expected gain after paying $1 to play.
d) Why do you think that you win something if you match 0 numbers, rather than say winning something if you match 5 numbers?