5. Carl really likes to go fishing, and he knows that he has a constant 32% chance of catching a fish each time he casts a
line. Carl is going fishing after work, but doesn't have a lot of time. He plans to only cast lines until he catches his first
fish of the evening, then he will go home.
a. Name the distribution that describes this situation.
b. How many lines can Carl expect to cast before he catches a fish?
c. What is the probability that Carl catches a fish on the first try?
d. Calculate the probability that will cast at least 8 lines before he catches a fish.
e. What is the probability that Carl catches a fish on his 4th cast?

a. The distribution that describes this situation is the geometric distribution. The geometric distribution models the number of trials needed to achieve the first success in a series of independent, identical trials, where each trial has a constant probability of success.

b. The expected number of lines Carl can expect to cast before he catches a fish is the reciprocal of the probability of catching a fish on each cast. In this case, Carl has a 32% chance of catching a fish on each cast, so the expected number of lines he needs to cast is 1 / 0.32 ≈ 3.125. Therefore, Carl can expect to cast approximately 3 lines before catching a fish.

c. The probability that Carl catches a fish on the first try is equal to the probability of success, which is 32% or 0.32.

d. To calculate the probability that Carl will cast at least 8 lines before catching a fish, we need to calculate the complement of the probability of catching a fish within 7 lines. The complement is 1 minus the probability of catching a fish within 7 lines.

P(casting at least 8 lines) = 1 - P(catching a fish within 7 lines)

Since each line has a 32% chance of success, the probability of not catching a fish in 7 lines is (1 - 0.32)^7 = 0.3231. Therefore, the probability of casting at least 8 lines before catching a fish is approximately 1 - 0.3231 = 0.6769, or 67.69%.

e. The probability that Carl catches a fish on his 4th cast is calculated as follows:

P(catching a fish on the 4th cast) = (1 - 0.32)^3 * 0.32 = 0.2176, or 21.76%. This is because the first three casts need to be unsuccessful (with a probability of (1 - 0.32)^3) and the fourth cast needs to be successful (with a probability of 0.32).