Final answer:
P (0 < x < 12) is calculated by multiplying the constant value of the continuous probability function f(x) = 12 by the width of the interval (12), and scaling the result to ensure the total area under the curve equals 1, which yields a probability of 1.
Step-by-step explanation:
Understanding Continuous Probability Functions
If we are given that f(x), a continuous probability function, equals to 12 across the interval 0 ≤ x ≤ 12, we can determine P (0 < x < 12). Since a probability function must integrate to 1 over its entire range, and given that f(x) is constant, the area under the curve (which represents the probability) can be found by simply multiplying the function's value by the width of the interval over which it is defined.
In this case, since f(x) = 12 across the interval from 0 to 12, the area of the rectangle formed by the function and the x-axis is 12 (height) × 12 (width). However, to satisfy the requirements of a probability function, this area must equal 1, which implies that the calculation for probability over any subinterval must consider the scaling factor applied to the function.
Thus, the probability that x falls between 0 and 12, P (0 < x < 12), is equal to the function's value (12) multiplied by the width of the interval (12), and then scaled appropriately to ensure the total area under the function equals 1. In practice, we find that the scaling factor is 1/144, making P (0 < x < 12) to be 1 once the correct scaling is accounted for.