Question

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day.

x 25 50 75 100 Total
P(x) .24 .33 .29 .14 1.00

(a) Find the probability expressions: (Round your answers to 2 decimal places.)
b. P(X ≤ 50)
c. P(X > 25)
d. P(X < 75)

Answer 1

The probabilities calculated for Jane's tube rentals are as follows: P(X ≤ 50) is 0.57, P(X > 25) is 0.76, and P(X < 75) is 0.57.

Step-by-step explanation:

To solve the given problems involving probability distributions, we will look at the cumulative probabilities and the complement rule. We'll start by calculating each required probability.

• b. P(X ≤ 50) is the probability that Jane rents 50 or fewer tubes. Given her probabilities P(25) = 0.24 and P(50) = 0.33, we add these to find the cumulative probability: P(X ≤ 50) = P(25) + P(50) = 0.24 + 0.33 = 0.57.
• c. P(X > 25) is the probability that Jane rents more than 25 tubes, which is the complement of P(X ≤ 25). Using the complement rule: P(X > 25) = 1 - P(X ≤ 25) = 1 - P(25) = 1 - 0.24 = 0.76.
• d. P(X < 75) is the probability that she rents fewer than 75 tubes. This includes the probabilities of renting 25 and 50 tubes. So, P(X < 75) = P(X ≤ 50) = 0.57 (as calculated in part b).

Answer 2

P(X ≤ 50) = 0.57

P(X > 25) = 0.76

P(X < 75) = 0.57

Step-by-step explanation:

Given: probability distribution to the number of tubes that will rent on a randomly selected day.

x ⇒ the number of tubes

P(x) ⇒ probability

x : 25 50 75 100 Total

P(x) : 0.24 0.33 0.29 0.14 1

So,

P(x ≤ 50) = P(x = 25) + P(x = 50) = 0.24 + 0.33 = 0.57

P(x > 25) = Total - P(x = 25) = 1 - 0.24 = 0.76

P(x < 75) = P(x = 25) + P(x = 50) = 0.24 + 0.33 = 0.57