Probability Theory
A commuter bus has \( 10 \) seats. The probability that any passenger will not show up for the bus is \( 0.6, \) independent of other passengers. If the bus company sells \( 12 \) tickets for the bus operation, what is the probability that at least one passenger will have to stand?
A parking building which is open for \(7\) hours a day has the following fee policy: \( 18 \) dollars per hour for the first 3 hours of parking, and \( 6 \) dollars for each additional hour. Many years of data shows that the number of hours of parking for a car, denoted \( X, \) is a discrete random variable with probability function \[ P(X = k) = \begin{cases} \frac{8 - k }{28}\ ( k = 1,2, \cdots, 7 ) \\ 0 \text{ otherwise.} \end{cases} \]
What is the expected parking charge for a car in dollars under this policy?
Let \( X_1 \) and \( X_2 \) be random samples from a discrete distribution with probability function \[ P(X=k) = \begin{cases} \frac{1}{2} \quad ( k = 0 ) \\ \frac{1}{2} \quad ( k = 1 ). \end{cases} \] What is the expected value of \( X_1 \times X_2? \)
South Kingston High School, where James is attending, has a policy of giving discipline at weekend to those who were late for school in that week more than \( 2 \) times. The probability that James is late for school is \( \frac{2}{13}. \) The tardiness that occurs in any given day is independent of the tardiness that occurs in other days. What is the probability that James gets disciplined this weekend?
Note: James goes to school 5 days a week from Monday through Friday.
Kate and Devin are playing a game with two six-sided dice, each numbered 1 to 6. They each toss a die at the same time. If the product of the two numbers on the two dice is bigger than or equal to \( 25, \) Devin wins the game. What is the probability that Devin wins the game?