Question

To pass the time, a toll booth collector counts the number of cars that pass through his booth until he encounters a driver with red hair. Suppose we define the random variable Y = the number of cars the collector counts until he gets a red-headed driver for the first time. Is Y a binomial random variable?

Answer 1

Y is not a binomial random variable but a geometric random variable, as it represents the number of trials until the first success in a series of independent Bernoulli trials with constant success probability.

Step-by-step explanation:

No, the random variable Y = the number of cars the collector counts until he gets a red-headed driver for the first time is not a binomial random variable, but rather a geometric random variable.

The main characteristics of a geometric random variable are that we are waiting for the first success in a series of Bernoulli trials, and the trials are independent with the same probability of success each time. In this case, each car passing through the toll is a trial, and finding a red-headed driver is defined as a success.

The probability of finding a red-headed driver remains the same for each car, and the outcomes of the trials do not affect one another.

The variable Y takes on the values 1, 2, 3, and so on, as the collector can count indefinitely until a red-headed driver appears.