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

- Being correct only on questions 1 and 4.
- Being correct on any two questions.

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.

- What type of random variable is \(X\)?
- Calculate the probability that
- \(P(X = 15), \text{ eg } f(15)\)
- \(P(X > 15)\)
- \(P(X \leq 15)\)

- Give values for the expected value, variance, and standard deviation.

- 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.
- Assume \(X \sim \text{Bernoulli}(p)\).
- Calculate the variance of a Bernoulli(p) random variable.
- Plot using
`bp.curve()`

the variance as a function of \(p\), across the domain \([0, 1]\). - Guess the value of \(p\) for which the variance is maximized.
- Use calculus to find the value of \(p\) for which the variance is maximized.