Final answer:
The random variable X represents the number of students out of a group of 50 who ride the bus to school. It follows a binomial distribution with a probability of success of 0.125. The expected number of students riding the bus is approximately 6 or 7. The probability of exactly 30 students riding the bus is approximately 4.37%.
Step-by-step explanation:
b. Values that the random variable X may take on: The random variable X represents the number of students out of a group of 50 who ride the bus to school. It can take on values from 0 (if none of the 50 students ride the bus) to 50 (if all 50 students ride the bus).
c. Distribution of X: The distribution of X is a binomial distribution, denoted as X ~ B(50, 0.125), where 50 is the number of trials (students) and 0.125 is the probability of success (students riding the bus).
d. Expected number of students riding the bus: To find the expected number of students riding the bus, we multiply the number of trials (50) by the probability of success (0.125): Expected value = 50 * 0.125 = 6.25. Therefore, we expect approximately 6 or 7 students to ride the bus.
e. Probability that exactly 30 students ride the bus: To find this probability, we use the binomial probability formula: P(X = 30) = (50 choose 30) * (0.125)^30 * (1 - 0.125)^(50 - 30). Calculating this using a calculator, we find that the probability is approximately 0.0437 or 4.37%.