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Chapter 6: Problem 63

Suppose that we define the following events: \(C=\) event that a randomly selected driver is observed to be using a cell phone, \(A=\) event that a randomly selected driver is observed driving a passenger automobile, \(V=\) event that a randomly selected driver is observed driving a van or SUV, and \(T=\) event that a randomly selected driver is observed driving a pickup truck. Based on the article "Three Percent of Drivers on Hand-Held Cell Phones at Any Given Time" (San Luis Obispo Tribune, July 24, 2001), the following probability estimates are reasonable: \(P(C)=.03\), \(P(C \mid A)=.026, P(C \mid V)=.048\), and \(P(C \mid T)=.019 .\) Ex- plain why \(P(C)\) is not just the average of the three given conditional probabilities.

Short Answer

Expert verified
The probability \(P(C)\) is not just the average of the three given conditional probabilities \(P(C \mid A)\), \(P(C \mid V)\), and \(P(C \mid T)\) because these probabilities don't represent the same conditions. The overall probability refers to any driver selected at random across all vehicle types, whereas the conditional probabilities refer to a specific subset of drivers who drive a specific type of vehicle. Hence, their average doesn't give the overall probability.

Step by step solution

01

Understanding Terms and Given Information

Let's understand the terms. The event 'C' represents the situation where a randomly selected driver is observed using a cellphone. The events 'A', 'V', and 'T' represent the observation of a driver driving a different type of vehicle, which are automobile, van/SUV, and pickup truck respectively. The probabilities of event C given A, V, and T have been given, along with the overall probability of C.
02

Comprehending Conditional Probability

Conditional probability, represented as \(P(C \mid A)\), \(P(C \mid V)\), and \(P(C \mid T)\), is the probability of an event (in this case, C) occurring given that another event (which are A, V, and T in this case) has already occurred. They are independent of each other and don't have an additive effect.
03

Understanding the Differences

The probability \(P(C)\) represents referring to any driver picked at random from the whole population, whereas the probabilities \(P(C \mid A)\), \(P(C \mid V)\), and \(P(C \mid T)\) represents conditional probabilities given a specific subset of the population (who drive a specific type of vehicle). Thus, the overall probability of a driver using a cellphone isn't just the average of these conditional probabilities as they represent different conditions.

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