Question
A degenerate gambler and a quantitative analyst go to the casino and play blackjack. The blackjack game has a \( 3\% \) edge per hand over the player. The gambler bets $50 per hand for 100 hands. The quantitative analyst bets that the gambler will not be profitable after 100 hands, and they both stake $50 on this side bet ($50 profit to the winner of the bet). What is the expected value of the degenerate gambler? Assume in blackjack there are no draws, only wins and losses.
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Solution
Answer: -165.57
We will first calculate the expected value of the 100 blackjack hands, and then calculate the expected value of the side bet.
The expected loss per blackjack hand is \( 50 * 0.03 = \$1.50 \). Therefore, the expected blackjack loss over 100 hands is: \[ 100 * \$1.50 = \$150 \] Now calculate the probability that the gambler is profitable after 100 hands. If each win gains $50 and each loss loses $50, then the win probability \( p \) implied by a 3% house edge satisfies: \[ -1.5 = 50p - 50(1-p) \] \[ p = 0.485 \] The gambler is profitable if he wins at least 51 of the 100 hands. Thus: \[ P(\text{profitable}) = P(X \ge 51), \quad X \sim B(100,0.485) \] \[ P(\text{profitable}) \approx 0.3443 \] The expected value of the side bet is: \[ E[\text{side bet}] = 0.3443 * 50 - (1 - 0.3443) * 50 \] \[ E[\text{side bet}] \approx -15.57 \] Therefore, the gambler's total expected value is: \[ -150 - 15.57 = -165.57 \]
The expected loss per blackjack hand is \( 50 * 0.03 = \$1.50 \). Therefore, the expected blackjack loss over 100 hands is: \[ 100 * \$1.50 = \$150 \] Now calculate the probability that the gambler is profitable after 100 hands. If each win gains $50 and each loss loses $50, then the win probability \( p \) implied by a 3% house edge satisfies: \[ -1.5 = 50p - 50(1-p) \] \[ p = 0.485 \] The gambler is profitable if he wins at least 51 of the 100 hands. Thus: \[ P(\text{profitable}) = P(X \ge 51), \quad X \sim B(100,0.485) \] \[ P(\text{profitable}) \approx 0.3443 \] The expected value of the side bet is: \[ E[\text{side bet}] = 0.3443 * 50 - (1 - 0.3443) * 50 \] \[ E[\text{side bet}] \approx -15.57 \] Therefore, the gambler's total expected value is: \[ -150 - 15.57 = -165.57 \]