In each hand of Blackjack, Megan wins $2 with probability 0.1, or wins $1 with probability 0.3, or loses $1 with probability 0.6. Suppose Megan continues to play 100 hands of Blackjack, and assume the outcomes of all plays are independent.
(a) Compute the expected payoff after 100 plays. (b) Approximate the probability that Megan will be winning after 100 plays.
To compute the expected payoff after 100 plays, we need to calculate the expected value or the average payoff per play and then multiply it by the number of plays.
Given the probabilities and corresponding payoffs, we can calculate the expected value per play:
Expected payoff per play = (0.1 * $2) + (0.3 * $1) + (0.6 * (-$1))
= $0.2 + $0.3 + (-$0.6)
= -$0.1
Therefore, the expected payoff per play is -$0.1 or a loss of $0.1.
To calculate the expected payoff after 100 plays, we simply multiply the expected payoff per play by the number of plays:
Expected payoff after 100 plays = -0.1 * 100
= -$10
Thus, the expected payoff after 100 plays is a loss of $10.
To approximate the probability that Megan will be winning after 100 plays, we can use the Central Limit Theorem. According to this theorem, when the number of independent, identically distributed random variables (plays in this case) is large, the sum of those variables will be approximately normally distributed.
Given that the probability distribution of each play is not symmetric, we cannot directly use the mean and standard deviation to approximate the probability.
However, we can use a simulation approach to estimate the probability. Here's how we can do it:
1. Write a computer program or use a simulation tool to simulate playing 100 hands of Blackjack, repeatedly. Randomly assign outcomes based on the given probabilities for each play.
2. Run the simulation for a large number of trials, such as 100,000.
3. Count the number of trials where Megan ends up with a positive total payoff after 100 plays.
4. Approximate the probability by dividing the count from step 3 by the total number of trials (100,000 in our example).
By using this simulation approach, we can estimate the probability that Megan will be winning after 100 plays.