1. Suppose an exam has 6 multiple-choice questions, and that each question has 5 possible answers. On each question, only 1 answer is correct. If a student guesses randomly and independently from question to question,find the probability of

    1. Being correct only on questions 1 and 4.
    2. Being correct on any two questions.
  2. It is claimed that 80% of American drivers engage in multitasking, driving while eg talking on their cell phone, eating a snack, or texting. In a random sample of 20 drivers, let \(X\) equal the number of multitaskers.

    1. What type of random variable is \(X\)?
    2. Calculate the probability that
      • \(P(X = 15), \text{ eg } f(15)\)
      • \(P(X > 15)\)
      • \(P(X \leq 15)\)
    3. Give values for the expected value, variance, and standard deviation.
  3. Assume the random variables \(X_n\) are independent, for \(n = 1, 2, 3, 4\), and that \(P(X_n = 1) = p_n\), where \(1\) indicates success. Calculate the probability of success of the entire system.
  4. Assume \(X \sim \text{Bernoulli}(p)\).
    1. Calculate the variance of a Bernoulli(p) random variable.
    2. Plot using bp.curve() the variance as a function of \(p\), across the domain \([0, 1]\).
    3. Guess the value of \(p\) for which the variance is maximized.
    4. Use calculus to find the value of \(p\) for which the variance is maximized.