A consumer looking to buy a used red Miata car will call...

A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call. a. In words, define the random variable $X.$ b. List the values that $X$ may take on. c. Give the distribution of $X . X \sim$ ____ (__,___) d. On average, how many dealerships would we expect her to have to call until she finds one that has the car? e. Find the probability that she must call at most four dealerships. f. Find the probability that she must call three or four dealerships.

A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call.
a. In words, define the random variable $X.$
b. List the values that $X$ may take on.
c. Give the distribution of $X . X \sim$ ____ (__,___)
d. On average, how many dealerships would we expect her to have to call until she finds one that has the car?
e. Find the probability that she must call at most four dealerships.
f. Find the probability that she must call three or four dealerships.

Key Concepts

Random Variable

A random variable assigns a numerical value to each outcome of a random process, allowing us to quantitatively study uncertainty. It functions as a bridge between abstract experimental outcomes and concrete mathematical analysis.

Expected Value

In a geometric distribution, the expected value is the average number of trials required to obtain the first success. It is calculated as the reciprocal of the probability of success, meaning that if the success probability is low, you would expect to conduct more trials on average.

Cumulative Distribution Function (CDF)

The cumulative distribution function of a discrete random variable, such as one following a geometric distribution, gives the probability that the variable's value is less than or equal to a certain number. It aggregates the probabilities of all outcomes up to that point, which is useful for determining the likelihood of an event occurring within a specified number of trials.

Probability Mass Function (PMF)

The probability mass function specifies the probability that a discrete random variable takes on a particular value. In the case of a geometric distribution, it provides the probability that the first success occurs on the nth trial by combining the probability of failures in all previous trials with the probability of success on the nth trial.

Geometric Distribution

The geometric distribution models the number of independent trials needed to achieve the first success in a series of Bernoulli (success/failure) experiments. Its key characteristic is that each trial has the same constant probability of success, and the trials are independent.

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