Question

[Marathon Blackjack] In a particular version of the card game blackjack offered by gambling casinos, if a player uses a particular optimized strategy, then in one game with one unit of money initial bet, the net return is -0.0029 and the standard deviation of the net return is 1.1418 . Suppose a player uses this strategy and bets $100 on each game, regardless of how much the player won or lost in previous games. (a) What is the probability the player is ahead by at least $1000 after 1000 games? (Use the central limit theorem

Answer 1

In Marathon Blackjack with an optimized strategy, after 1000 games at $100 per bet, the player's probability of being ahead by at least $1000 can be calculated using the Central Limit Theorem, which requires computing a z-score and consulting a z-table or statistical software.

Step-by-step explanation:

Probability and Expected Value in Games of Chance

When analyzing the game of Marathon Blackjack, we see that the expected net return per game is -0.0029, with a standard deviation of the net return being 1.1418. Betting $100 per game, the question asks for the probability of being ahead by at least $1000 after 1000 games, applying the Central Limit Theorem (CLT). The CLT tells us that as the number of games increases, the distribution of total earnings will approach a normal distribution with a mean equal to the sum of expected returns and a standard deviation equal to the square root of the sum of variances. For 1000 games, the new expected return is 1000 times -0.0029 times $100, which is $-290, and the new standard deviation is sqrt(1000) times 1.1418 times $100, which is approximately $36037.70.

To find the probability of being ahead by at least $1000, we need to calculate the z-score for $1000 with this new distribution. Subtract the mean from $1000 and divide by the new standard deviation to find the z-score. Look up the cumulative probability for this z-score in a z-table or use a statistical software, to determine the probability of being ahead. Generally, as this is a negative expectation game, it is unlikely the player will be ahead by this margin after 1000 games.