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Chapter 3: Problem 70

A toll bridge charges \(\$ 1.00\) for passenger cars and \(\$ 2.50\) for other vehicles. Suppose that during daytime hours, \(60 \%\) of all vehicles are passenger cars. If 25 vehicles cross the bridge during a particular daytime period, what is the resulting expected toll revenue? [Hint: Let \(X=\) the number of passenger cars; then the toll revenue \(h(X)\) is a linear function of \(X\).]

Short Answer

Expert verified
The expected toll revenue is $40.00.

Step by step solution

01

Define Variables

Identify the variables involved in the problem. Let \( X \) represent the number of passenger cars crossing the bridge. Therefore, the number of other vehicles crossing the bridge would be \( 25 - X \).
02

Determine Probabilities

Since \( 60\% \) of all vehicles are passenger cars, the probability that a vehicle is a passenger car \( P(passenger\ car) = 0.6 \). Thus, \( X \) follows a binomial distribution with parameters \( n = 25 \) and \( p = 0.6 \).
03

Establish Revenue Function

The toll revenue function is determined by the number of passenger and other vehicles: \[ h(X) = 1.00 \times X + 2.50 \times (25 - X) \] Simplify this function to express the revenue in terms of \( X \): \[ h(X) = X + 62.5 - 2.5X = 62.5 - 1.5X \].
04

Calculate Expected Value

Compute the expected value of \( X \), \( E(X) \), using the binomial distribution: \[ E(X) = n \times p = 25 \times 0.6 = 15 \].
05

Substitute and Calculate Expected Revenue

Substitute \( E(X) = 15 \) into the revenue function to find the expected toll revenue: \[ E(h(X)) = 62.5 - 1.5 \times E(X) = 62.5 - 1.5 \times 15 = 62.5 - 22.5 = 40 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Binomial Distribution
A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states under a given number of observations. In our example of the toll bridge, we consider each vehicle crossing as a trial. It can either be a passenger car or another type of vehicle.
Here are some key characteristics of binomial distribution:
  • It involves a fixed number of trials, denoted as \( n \). In our case, this is 25 vehicles.
  • Each trial has two possible outcomes, like being a passenger car or not.
  • The probability of each outcome is constant; for passenger cars, it's 60% or 0.6.
  • One outcome is considered a success, let's say, identifying a passenger car.
Using these properties, we can model the number of passenger cars (\( X \)) as a binomial random variable, which helps in easily calculating the expected outcome.
Probability and Its Role
Probability measures the likelihood of a particular event occurring. In our task, the event is whether a vehicle crossing the bridge is a passenger car. This probability is given as 0.6, meaning there's a 60% chance that any given vehicle is a passenger car.
When dealing with probability in statistics or real-world scenarios like this, we rely on specific rules:
  • Probabilities range from 0 to 1, where 0 means the event never occurs, and 1 means it always occurs.
  • The probability of all possible outcomes must sum to 1. So, if there's a 60% chance the vehicle is a passenger car, there's a 40% chance it is not.
  • Using these probabilities, we can calculate expected values, like the expected number of passenger cars or toll revenue.
In our context, these probabilities enable us to apply the binomial distribution to predict the expected number of passenger cars.
Revenue Function in Practice
A revenue function is a mathematical expression that models how much money is generated under certain conditions. In the toll bridge problem, the revenue depends on the number of cars and their type.
Here鈥檚 how we define the revenue function:
  • Passenger cars pay \(1.00 each, and other vehicles pay \)2.50 each.
  • If \( X \) is the number of passenger cars, then \( 25 - X \) is the number of other vehicles.
  • The revenue function then combines these to give: \( h(X) = 1.00 \times X + 2.50 \times (25 - X) \).
The function simplifies to \( h(X) = 62.5 - 1.5X \), showing how revenue decreases as the count of passenger cars increases, given the higher charge for other vehicles.
This function is instrumental when calculating expected revenue, as it allows substitution with expected values, such as \( E(X) \), to predict outcomes and plan financially.

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