Problem 1. When P(A) = 1/3, P(B) = 1/2 and P(A U B) = 3/4 what is
a. P(A ∩ B)
b. P(Aᶜ U Bᶜ)

Problem 3. Consider the example of a chemical reaction that takes place in a reactor vessel. Assume that the process works as follows: On one side fluid or gas flows in, mixes with whatever is already present in the vessel, and eventually flows out on the other side. The length of time particles stay inside the vessel before moving on is called the residence time. Let At denote the event that the particle has a residence time longer than t seconds. Correspondingly Assume P(At) = e⁻ᵗ. Determine P(A₃ | A₄).

Problem 4. A certain grapefruit variety is grown in 2 regions in southerns Spain. Both areas get
infested from time to time with parasites that damage the crop. Let A be the event that region R₁
is infested with parasites and B that region R₂. Suppose P(A) = 3/4, P(B) = 2/5 and P(AUB) = 4/5. If the food inspection detects the parasite in a ship carrying grapefruits from R₁, what is the
probability that region R₂ is infested as well.

Problem 5. Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once out of every 200 packages; Tom who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeff who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of film does not show the expiration date, what is the probability that it was inspected by John?

Problem 6. Suppose the probability density function for the continuous random variable X is defined as follows: f(x) = 4x - 4x³, 0 ≤ x ≤ 1 and 0 elsewhere
1. Determine the expected value of the random variable 2X + 3
2. Determine the variance of the random variable 2X + 3.

Problem 7. Let a continuous random variable X be given that takes values in [0, 1] and whose distribution function F satisfies F(x) = 2x² - x⁴ for 0 ≤ x ≤ 1
1. What is the probability density function of X ?
2. Compute the probability P(1/4 ≤ x ≤ 3/4).

Problem 1. When P(A) = 1/3, P(B) = 1/2 and P(A U B) = 3/4 what is...