On average, 30-minute television sitcoms have 22 minutes of programming.If a probability distribution for minutes of programming can be approximated by a uniform distribution from 18 minutes to 26 minutes. What is the probability a sitcom will have 25 or more minutes of programming?
To find the probability that a sitcom will have 25 or more minutes of programming, we can start by calculating the probability of having less than 25 minutes of programming and then subtract it from 1 to get the complementary probability.
In a uniform distribution, the probability density function (PDF) is constant within the range of the distribution and 0 outside of it. So, the PDF for this distribution is:
f(x) = 1/(26-18) = 1/8
To calculate the probability of having less than 25 minutes of programming, we need to find the area under the PDF curve from 18 to 25. Since the PDF is a rectangle, the area is given by the width of the interval (25-18 = 7) times the height of the rectangle (1/8):
P(X < 25) = (25 - 18) * (1/8) = 7/8
Finally, we can subtract this probability from 1 to find the probability of having 25 or more minutes of programming:
P(X ≥ 25) = 1 - P(X < 25) = 1 - 7/8 = 1/8
Therefore, the probability that a sitcom will have 25 or more minutes of programming is 1/8 or 0.125.