Questions: You play a game of chance that you can either win or lose (there are no other possibilities) until you lose. Your probability of losing is p=0.57. What is the probability that it a. takes 4 games until you lose. b. takes 7 games until you lose. c. Find the mean c. Find the standard deviation.

Solution Steps

To solve this problem, we need to use the geometric distribution, which models the number of trials until the first success (or failure in this case). The probability mass function (PMF) for a geometric distribution is given by \( P(X = k) = (1-p)^{k-1} \cdot p \), where \( p \) is the probability of failure and \( k \) is the number of trials.

a. For the probability that it takes 4 games until you lose, we use the PMF with \( k = 4 \).

b. For the probability that it takes 7 games until you lose, we use the PMF with \( k = 7 \).

c. The mean of a geometric distribution is given by \( \frac{1}{p} \).

d. The standard deviation of a geometric distribution is given by \( \sqrt{\frac{1-p}{p^2}} \).

To find the probability that it takes 4 games until you lose, we use the geometric distribution formula: \[ P(X = k) = (1-p)^{k-1} \cdot p \] Given \( p = 0.57 \) and \( k = 4 \): \[ P(X = 4) = (1-0.57)^{4-1} \cdot 0.57 = 0.04532 \]

To find the probability that it takes 7 games until you lose, we use the same geometric distribution formula: \[ P(X = k) = (1-p)^{k-1} \cdot p \] Given \( p = 0.57 \) and \( k = 7 \): \[ P(X = 7) = (1-0.57)^{7-1} \cdot 0.57 = 0.003603 \]

The mean of a geometric distribution is given by: \[ \mu = \frac{1}{p} \] Given \( p = 0.57 \): \[ \mu = \frac{1}{0.57} = 1.754 \]

The standard deviation of a geometric distribution is given by: \[ \sigma = \sqrt{\frac{1-p}{p^2}} \] Given \( p = 0.57 \): \[ \sigma = \sqrt{\frac{1-0.57}{0.57^2}} = 1.150 \]

Final Answer