Final answer:
To find the probability P(0 < x < 12) for a uniform distribution function f(x) constant over the interval [0, 12], we calculate the area under the curve, which is 1, indicating certainty within that range. For P(0 < x < 4) with f(x) = 1/10, the probability is 0.4 as it is the product of interval length and f(x)'s height.
Step-by-step explanation:
Continuous Probability Functions and Probabilities
When dealing with continuous probability functions, such as f(x), understanding how to find the probability within a certain range is crucial. For a uniform distribution, where the function f(x) is constant across an interval, calculating the probability is straightforward since each outcome within the interval is equally likely.
Considering the case where f(x) is defined as 12 for the interval 0 ≤ x ≤ 12, the function describes a uniform distribution. To find the probability P(0 < x < 12), we note that the area under the curve of f(x) across this interval represents the probability of the event occurring within that range. Since the function is constant, the area and thus the probability of being within the interval (0, 12) is 1, signifying certainty, as the value of x is confined to be within those bounds due to the definition of the function.
In the case of f(x) = 1/10 for 0 ≤ x ≤ 10, finding P(0 < x < 4) involves calculating the area under the curve of f(x) between x=0 and x=4. Since f(x) is constant, this probability equals the length of the interval (4 - 0) times the height of f(x) (1/10), which results in 4/10 or 0.4.