questions

Below are some new practice questions covering random variables. Try solving each one using what we discussed!

Question 1: Bernoulli Random Variable

Imagine a factory produces light bulbs that have a 95% chance of being good (success) and a 5% chance of being defective (failure).

• Task:
  • What is the probability that a randomly chosen bulb is defective?
  • (Bonus: If we define “good” as 1 and “defective” as 0, what is the expected value of this Bernoulli random variable?)

Hint: A Bernoulli random variable only takes two values (0 and 1) and its probability function is:
  f(x) = p when x = 1 and 1 – p when x = 0.

Question 2: Binomial Distribution

A student takes a multiple-choice quiz with 10 questions. Each question has 4 choices (only one is correct), and the student guesses on every question.

• Task: What is the probability that the student answers exactly 3 questions correctly?

Hint: Use the binomial formula:
  P(X = x) = (n choose x) * (p)^x * (1 – p)^(n – x)
Here, n = 10 and p = 0.25.

Question 3: Geometric Distribution

A basketball player has a 40% chance of making a free throw on any given attempt.

• Task: What is the probability that her first successful free throw happens on her 4th attempt?

Hint: For a geometric random variable, the probability that the first success occurs on the n-th trial is:
  P(X = n) = p * (1 – p)^(n – 1).

Question 4: Poisson Distribution

A call center receives an average of 2 calls per minute.

• Task: What is the probability that exactly 3 calls are received in a given minute?

Hint: The Poisson probability mass function is:
  P(X = x) = (λ^x / x!) * e^(–λ)
where λ is the average rate (λ = 2 in this case).

Question 5: Uniform Distribution

A number is chosen at random from the interval [0, 10] with every number equally likely.

• Task: What is the probability that the chosen number lies between 3 and 7?

Hint: For a uniform distribution on [a, b], the probability that the number falls between c and d (with a ≤ c < d ≤ b) is:
  Probability = (d – c) / (b – a).

After you work through these, we can compare how these new questions relate to the counting and sampling ideas from Lecture 4. They build on the idea of assigning numbers (or “scores”) to outcomes and calculating the chance of those numbers occurring. Let me know when you’re ready to discuss your answers or if you need help with any of them!