A couple has agreed to attend a “casino night” as part of a fundraiser for the local hospital, but they believe that gambling is generally a losing proposition. For the sake of the charity, they have decided to attend and to allocate $300 for the games. There are to be four games, each involving standard decks of cards.
The first game, called Jack in 52, is won by selecting a Jack of a specific suit from the deck. The probability of actually doing this is, of course, 1 in 52 (= 0.0192). Gamblers may place a bet of $1, $2, or $4 on this game. If they win, the payouts are $12 for a $1 bet, $24.55 for a $2 bet, and $49 for a $4 bet.
The second game, called Red Face in 52, is won by selecting from the deck a red face card (i.e., red Jack, red Queen, or red King). The probability of winning is 6 in 52 (= 0.1154). 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, called Face in 52, is won by selecting one of the 12 face cards from the deck. The probability of winning is 12 in 52 (= 0.2308). Payouts are $4, $8.15, and $16 for $1, $2, and $4 bets.
The last game, called Red in 52, is won by selecting a red card from the deck. The probability of winning is 26 in 52 (= 0.50). Payouts are $1.80, $3.80, and $7.50 for $1, $2, and $4 bets.
Given that they can calculate the expected return (or, more appropriately, loss) for each type of game and level of wager, they have decided to see if they can minimize their total expected loss by planning their evening using LP. For example, the expected return from a $1 bet in the game Jack in 52 is equal to $0.2308 (= $12 X 1/52 + $0 X 51/52). Since the amount bet is $1, the expected loss is equal to $0.7692 (= $1 - $0.2308). All other expected losses can be calculated in a similar manner.
They want to appear to be sociable and not as if they are trying to lose as little as possible. 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 bets, at least $50 on $2 bets, and at least $72 on $4 bets. They will bet no more than (and no less than) the agreed-upon $300. What should be their gambling plan, and what is their expected loss for the evening?