A popular casino in Windsor estimates that the chance that someone sitting at one of their
Black Jack tables has a 0.35 probability of winning a given hand. Assume that all games
played are independent.
a. Isaac decides to play until his first win, and bets $50 per game. When he wins, he
will win $100. What is Isaac’s expected net gain/loss from this strategy?
b. Suppose the casino has 100 people playing the same strategy as Isaac. What is the
probability that the casino sees no net gain from these 100 people? Hint: recall that
the variance of a geometric random variable is (1 p)=p2:
I am not sure how to find the Variance for part b
Thank you
To find the expected net gain/loss for Isaac's strategy, we need to calculate the expected amount he will win on each game.
a. Isaac has a 0.35 probability of winning a given hand, and when he wins, he will win $100. So his expected amount won from each game can be calculated as follows:
Expected amount won = (Probability of winning) * (Amount won when winning) + (Probability of losing) * (Amount lost when losing)
= (0.35) * ($100) + (0.65) * (-$50)
= $35 - $32.5
= $2.5
Therefore, Isaac's expected net gain/loss from this strategy can be calculated as follows:
Expected net gain/loss = (Expected amount won) * (Number of games played)
= ($2.5) * (∞) [since Isaac plays until his first win]
= ∞
b. To find the probability that the casino sees no net gain from 100 people playing the same strategy as Isaac, we can use the concept of a geometric distribution. In our case, the probability of winning a given hand is 0.35, so the probability of losing a given hand is 0.65.
The variance of a geometric random variable can be calculated using the formula: (1 - p) / (p^2), where p is the probability of success.
Variance = (1 - p) / (p^2)
= (1 - 0.35) / (0.35^2)
= (0.65) / (0.1225)
= 5.30612245
Now, as we have 100 people playing the same strategy, we can use the concept of the binomial distribution to find the probability that the casino sees no net gain from these 100 people. The binomial distribution formula is:
Probability (no net gain) = C(n, k) * (p^k) * ((1 - p)^(n - k))
where n is the number of trials (100 people), k is the number of successful trials (0 net gain), and p is the probability of success (0.35).
Plugging in the values, we get:
Probability (no net gain) = C(100, 0) * (0.35^0) * ((1 - 0.35)^(100 - 0))
= 1 * 1 * (0.65^100) [as C(100, 0) = 1 and any number raised to the power of 0 is 1]
= 1 * 1 * (0.021283...)
= 0.021283...
Therefore, the probability that the casino sees no net gain from these 100 people is approximately 0.0213, or 2.13%.