Based on Problems 1 through 4 from the provided document, here is a similar set of problems:

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Problem 1:
Three events $A$, $B$, and $D$ satisfy the following probabilities:

1. $P(A^c) = 0.25$
2. $P(B) = 0.6$
3. $P(D) = 0.3$
4. $P(B \cup D) = 0.7$
5. $P(A \cap B) = 0.45$

a) Find $P(B \cap D)$.
b) Calculate $P(D^c | A)$.
c) Determine if events $A$ and $D$ are independent.

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Problem 2:
For a series of training sessions, say two out of five sessions are led by Sam, and the rest are led by Taylor. The errors made by each instructor during a session are as follows:

• The number of errors Sam makes in a session is a uniform discrete random variable with range $[0, 2]$.
• The number of errors Taylor makes in a session is a Poisson random variable with a mean of 1.

If a session is selected at random:

a) What is the probability that the instructor makes no errors in the session?
b) If exactly one error occurs in the session, what is the probability that the instructor is Sam?

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Problem 3:
Five marbles are drawn randomly from a bag containing 50 marbles, where each marble is one of four colors: red, blue, green, or yellow. There are 13 marbles of each color, and each color is equally likely.

a) What is the probability that the draw contains no red or blue marbles?
b) What is the probability that exactly one marble is red and one marble is blue?

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Problem 4:
Let $X$ and $Y$ be two random variables:

• $X$ is a normal (Gaussian) random variable with a mean of 0 and a variance of 9.

• $Y$ is a geometric random variable with success probability $\frac{1}{4}$.

  a) What is $P(X > 2)$?
  b) Calculate the expected value of the new random variable $Z = X + 3Y$.
  c) Given $X$ and $Y$ are independent, find the variance of the new random variable $U = 4X - Y$.

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Would you like to go over the solutions or explore any of these problems in further detail?

Additional Related Questions

1. How do you calculate the probability of events being independent?
2. What are the steps to compute the expected value of a sum of random variables?
3. How does the uniform distribution differ between discrete and continuous cases?
4. What’s the effect of changing the mean of a normal distribution on probability calculations?
5. In what situations would you use a geometric distribution in real-life scenarios?

Tip:

When determining independence between two events, verify if $P(A \cap B) = P(A) \times P(B)$. This is the mathematical condition for independence.