Using the law of conditional probabilities, calculate the probability of the player winning.

Let’s return to the casino game Red Dog from Problem 3.25. Using the law of conditional probabilities, calculate the probability of the player winning. Problem 3.25 Red Dog is a casino game played with a deck of 52 cards. Suit plays no role in determining the value of each card. An ace is worth 14, king 13, queen 12, jack 11, and numbered cards are worth the number indicated on the card. After staking a bet a player is dealt two cards. If these two cards have a “spread” of one or more, a third card is dealt. The spread is defined as the number of points between the values of the two cards dealt (e.g., if a player is dealt a 5 and a 9, he has a spread of three). When a player has a spread of at least one, he may choose to double his initial stake before the third card is dealt. At this point, the third card is dealt. If the value of the third card lies between the two cards dealt earlier, the player gets a payoff of s times his final stake plus the final stake itself, where s = 5 for a spread of 1, s = 4 for a spread of 2, s = 2 for a spread of 3, and s = 1 for a spread of 4 or more. In cases where the value of the two cards dealt is sequential (e.g., 7 and 8), no third card is dealt and the player gets his initial stake back. If the values of the two cards dealt are equal, the player immediately gets a third card. If this third card has the same value as the other two, the player gets a payoff of 11 times his initial stake plus the stake itself. The player applies the following simple strategy. The initial stake is only doubled if the spread equals 7 or more. Can you explain why it is not rational to double the stake if the spread is less than 7? Using computer simulation, determine the house percentage for Red Dog.

 

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