A long-distance taxi service owns four vehicles. These are...

A long-distance taxi service owns four vehicles. These are of different ages and have different repair records. The probabilities that, on any given day, each vehicle will be available for use are $0.95,0.90,0.90$, and $0.80$ . Whether one vehicle is available is independent of whether any other vehicle is available. a. Find the probability distribution for the number of vehicles available for use on a given day. b. Find the expected number of vehicles available for use on a given day. c. Find the standard deviation of the number of vehicles available for use on a given day.

A long-distance taxi service owns four vehicles. These are of different ages and have different repair records. The probabilities that, on any given day, each vehicle will be available for use are $0.95,0.90,0.90$, and $0.80$ .
Whether one vehicle is available is independent of whether any other vehicle is available.
a. Find the probability distribution for the number of vehicles available for use on a given day.
b. Find the expected number of vehicles available for use on a given day.
c. Find the standard deviation of the number of vehicles available for use on a given day.

Key Concepts

Independence

Independence in probability means that the occurrence (or outcome) of one event does not affect the outcome of another. In the context of multiple Bernoulli trials, this property allows the joint probability of specific outcomes to be calculated as the product of the individual probabilities, simplifying the analysis of the overall distribution.

Variance and Standard Deviation

The variance of a sum of independent Bernoulli trials is found by summing the variances of the individual trials, where each variance is given by p(1 - p) for a trial with success probability p. The standard deviation, which measures the spread or dispersion of the distribution, is then the square root of the total variance.

Expected Value

The expected value (or mean) of a sum of independent Bernoulli random variables is the sum of their individual probabilities. This property, based on the linearity of expectation, allows the average number of successful events to be calculated easily regardless of the differences in individual probabilities.

Poisson Binomial Distribution

This distribution describes the sum of independent Bernoulli trials that do not necessarily have the same success probability. It is used when each individual event (e.g., a trial such as a vehicle being available) has a unique probability of success, and the overall distribution is determined by the different probabilities across the trials.

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