Consider the situation of Example 2.32 on page 78. (a) What...

Consider the situation of Example 2.32 on page 78.
(a) What is the probability that no more than 4 cars will be serviced by the mechanic?
(b) What is the probability that he will service fewer than 8 cars?
(c) What is the probability that he will service either 3 or 4 cars?

Key Concepts

Poisson Distribution

The Poisson distribution is a common probability distribution used to model the number of events occurring within a fixed interval of time or space when these events occur with a known constant rate and independently of the time since the last event. It is particularly applicable in contexts where the events are rare or random in nature, such as the number of cars serviced by a mechanic in a given time period.

Discrete Random Variables

A discrete random variable is one that takes on countable values, with each value having an associated probability. This concept is central in probability theory as it allows us to model scenarios where outcomes are distinct and separate, such as the number of cars serviced in a specific time period.

Probability Mass Function (PMF)

The probability mass function assigns probabilities to each possible outcome of a discrete random variable. It provides a formula or set of values that indicate the likelihood of observing a particular outcome, and is used to determine probabilities for events such as servicing exactly 3 or 4 cars.

Cumulative Distribution Function (CDF)

The cumulative distribution function gives the probability that a discrete random variable will take on a value less than or equal to a certain number. It is useful for computing the probability that a random variable falls within a specific range, such as no more than 4 cars or fewer than 8 cars serviced.

Summation of Probabilities

Summing the individual probabilities provided by the PMF is a standard method to compute the probability of a range of outcomes. For example, to find the probability that the number of serviced cars is within a specific range or equals one of several values, one sums the PMF values for those outcomes.

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