<p>In the popular lottery game \"Powerball\", players are asked to select 5 numb
ID: 2960390 • Letter: #
Question
<p>In the popular lottery game "Powerball", players are asked to select 5 numbered blue balls from a total of 59 numbered blue balls (order is irrelevant), and 1 numbered orange ball from the total of 39 numbered orange balls. The player wins money if some or all of the balls that the player selects match the winning balls according to the following:</p><p>Players Selected balls match then they win</p>
<p>all 5 blue and the 1 orange then they win the Jackpot.  All 5 blue but not the 1 orange then they win $200,000.  4 out of 5 blue and the 1 orange then they win $10,000.  4 out of 5 blue but not the 1 orange then they win $100.  3 out of 5 blue and the 1 orange then they win $100.  3 out of 5 blue but not the 1 orange then they win $7.  2 out of the 5 blue and the 1 orange then they win $7.  1 out of the 5 blue and the 1 orange then they win $4.  0 out of the 5 blue and the 1 orange then they win $3.    </p>
<p>Suppose the Jackpot is worh $100,000,000;  What is the players expected winnings for each time played? ( Assume that anyone who wins the Jackpot will win the entire amount)       </p>
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Explanation / Answer
To win the jackpot you need to get them all and the powerball
5/59*4/58*3/57*2/56*1/55*1/39
To win the $200,000 you need to miss the powerball, that shouldn't be too hard but we need to put it in (it is 38/39) so this makes 5/59*4/58*3/57*2/56*1/55*38/39
To win $10,000 you need (5)*5/59*4/58*3/57*2/56*54/55*1/39 The factor of five is there since we can miss a winning ball at any of the five times a ball is drawn. It is easy to pick a losing ball, but we still put it in 54/55
To win $100 the first way, we need (5)*5/59*4/58*3/57*2/56*54/55*38/39 to win it the second way we need two losing balls. We could pick the first losing ball in any of the five picks and the second losing ball in any of the remaining picks. This would give us 5*4 arrangments, but we can't tell losing balls apart so we need to divide by 2 5*4/2 = 10 This gives (10)*5/59*4/58*3/57*54/56*53/55*1/39 .
To win $7 the first way we need (10)*5/59*4/58*3/57*54/56*53/55*38/39 (putting 2 loosing in five picks is the same as putting 2 winning in five picks, so multiply by 10)
To win $7 the other way we need (10)*5/59*4/58*54/57*53/56*52/55*1/39 .
To win $4 we need (5)*5/59*54/58*53/57*52/56*51/55*1/39. To win $3 we need 54*53*52*51*50/(59*58*57*56*55)*1/39
Now we multiply the odds by what we would win and add them all together.
Jackpot odds times jackpot =.0042
200K times $200,000 odds = .0003
10K times $10,000 odds = .014
100 times $100 odds = .005 + .007
7 times $7 odds = .019 + .009
4 times $4 odds = .03
3 times $3 odds = .048
total 0.14 expect to win 14 cents, and most of this is from the single digit winnings. In other words we expect to lose 86 cents every time we play if it costs $1 to play
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