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The concept of expected value is pivotal in probability theory and statistics. It gives us the average or mean outcome we can anticipate from a particular random event if we could repeat it many times. In our toll bridge exercise, we identify the expected value by calculating how many passenger cars would typically cross during the daytime period.

To find this expected value, we take the total number of vehicles, which is 25, and multiply it by the probability that a vehicle is a passenger car, which is 60% or 0.6. Mathematically, this is expressed as:

• \( E(X) = 25 \times 0.6 = 15 \)

Thus, we expect on average to have 15 passenger cars in this period.

A linear function is an essential concept in algebra and calculus. It describes a relationship in which the change in the output is directly proportional to the change in the input. In the context of the toll revenue calculation exercise, the function for toll revenue is linear because the revenue changes at a constant rate with respect to the number of passenger cars.

Here, the toll revenue can be expressed as a linear function, \( h(X) \), where \( X \) represents the number of passenger cars, so:

• For each passenger car crossing the bridge, \( \\(1.00 \) is added to the revenue.
• For the other types of vehicles, \( \\)2.50 \) is added per vehicle.

Thus, the revenue function illustrates how changes in \( X \) (the number of passenger cars) affect the total revenue.

A probability distribution shows how the probabilities are distributed over the possible values of a random variable. In the toll revenue task, while the exercise simplifies the probability distribution by providing a direct expected value, typically you would examine the distribution of various numbers of passenger cars that might cross.

Since each type of vehicle crossing has an associated probability (60% for passenger cars and 40% for other vehicles), this determines how likely each specific configuration of vehicles is. This probability framework helps to compute expected values and analyze vehicle patterns on larger time scales.

Understanding this distribution is key to estimating revenues and making data-driven forecasts and decisions.

Calculating the expected toll revenue involves combining the revenues from both passenger cars and other vehicles. Once we've predicted the expected number of each vehicle type, calculating the revenue becomes straightforward by multiplying the number of each type by their respective tolls.

For passenger cars, with an expected 15 cars at \( \\(1.00 \) each, we get:

• \( 15 \times 1.00 = \\)15.00 \)

Other vehicles, with 10 expected and charged \( \\(2.50 \) each, result in:

• \( 10 \times 2.50 = \\)25.00 \)

Adding these gives the total expected toll revenue:

• \( 15.00 + 25.00 = \$40.00 \)

This calculation helps bridge operators understand the anticipated revenue during specific times and make necessary planning and budgeting decisions.