Linear Programming

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A couple has agreed to attend a “Casino Night” as part of a fundraiser for the local hospital. They do not like to gamble because they believe that gambling is generally a losing proposition. However, for the sake of the charity, they have decided to attend and spend $300 on the games. There will be four games, each involving standard decks of cards.
The first game, Jack in 52, is won if you select the Jack of a particular suit from the deck. The probability of actually doing this is 4 in 52 (4/52 or .0769). Gamblers may place bets of $1, $2, or $4 on this game. If they win, the payouts are $12.00 for a $1 bet, $24.55 for a $2 bet, and $49 for a $4 bet.
The second game, Red Face in 52, is won if you select a red face card (including the Jack, Queen, or King) from the deck. The probability of winning is 6 in 52 (.1154). Again, bets may be placed in denominations of $1, $2, and $4. Payouts are $8.10, $16.35, and $32.50, respectively.
The third game, Face in 52, is won if you select 1 of the 12 face cards from the deck. The probability of winning is 12 in 52 (.2308). Payouts are $4, $8.15, and $16 for $1, $2, and $4 bets.
The last game, Red in 52, is won if you select a red card from the deck. The probability of winning is 26 in 52 (.5). Payouts are $1.80, $3.80, and $7.50 for $1, $2, and $4 bets.
Given that they can calculate the expected return or loss for each type of game and level of wager, they have decided to see if they can minimize their expected loss by planning their evening using LP. For example, a $1 bet in Jack in 52 has a return of $12.00, but there is only a 1 in 13 chance of winning. Therefore, the expected value of the dollar bet is ($12.00*(1/13)) or $.9231. This computes to an expected loss of $1-.9231, or $.0769.
The couple wants to appear generous. Therefore, they will place at least 20 bets (of any value) on each of the four games. Further, they will spend at least $26 on 1-dollar bets, at least $50 on 2-dollar bets, and at least $72 on 4-dollar bets. They will bet exactly the agreed-upon $300. What should be their gambling plan, and what is their expected loss for the evening?

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