Question

In a particular version of the card game blackjack a player has a particular optimized strategy. In one game with one unit of money (1$) ,initial bet, the net return is -0.0029 and the standard deviation of the net return is 1.1418. A player uses this strategy and bets $100 on each game, regardless of how the player won or lost in previous games. The excepted net game with a bet of $100 on each game is -0.29 and a standard deviation of (1.148)(100) = 114.18. The player plays 1000 games with a total bet of ($100)(1000). Determine the probability the player is ahead by at least $1000 after 1000 games using the CLT.

Answer 1

To determine the probability that the player is ahead by at least $1000 after 1000 games using the Central Limit Theorem (CLT), calculate the z-score and use the standard normal distribution table. Find the value in the table that is closest to the z-score and subtract it from 1. This will give you the probability of being ahead by at least $1000 after 1000 games.

Step-by-step explanation:

To determine the probability that the player is ahead by at least $1000 after 1000 games using the Central Limit Theorem (CLT), we need to calculate the z-score and use the standard normal distribution table. The formula for calculating the z-score is:

z = (X - μ) / (σ / √n)

Where X is the desired value, μ is the sample mean, σ is the sample standard deviation, and n is the sample size. In this case, X = $1000, μ = -$29 (expected net gain), σ = $114.18 (standard deviation), and n = 1000.

Using the z-score formula, we can calculate the z-score:

z = (1000 - (-29)) / (114.18 / √1000)

Once we have the z-score, we can use the standard normal distribution table to find the probability. The probability of being ahead by at least $1000 is equal to 1 minus the cumulative probability of being below $1000. Find the value in the table that is closest to the z-score and subtract it from 1. This will give you the probability of being ahead by at least $1000 after 1000 games.