a statistics professor plans classes so carefully that the lenghts of her classes are uniformly distributed between 46.0 and 56.0 minutes. find the probability that a given class period runs less than 50.5 minutes
To find the probability that a given class period runs less than 50.5 minutes, we can use the probability density function (PDF) of the uniform distribution.
The uniform distribution is defined by two parameters: the minimum value (a) and the maximum value (b). In this case, a = 46.0 minutes and b = 56.0 minutes.
The PDF of the uniform distribution is given by:
f(x) = 1 / (b - a)
To find the probability that a random variable (in this case, the length of a class period) is less than a given value (50.5 minutes), we integrate the PDF from the minimum value to the given value:
P(X < 50.5) = integral[a to 50.5] (1 / (b - a)) dx
Substituting the values, we have:
P(X < 50.5) = integral[46.0 to 50.5] (1 / (56.0 - 46.0)) dx
Simplifying further:
P(X < 50.5) = (1 / 10) * integral[46.0 to 50.5] dx
Integrating, we get:
P(X < 50.5) = (1 / 10) * [x] (from 46.0 to 50.5)
Evaluating the definite integral, we have:
P(X < 50.5) = (1 / 10) * (50.5 - 46.0)
P(X < 50.5) = 0.45
Therefore, the probability that a given class period runs less than 50.5 minutes is 0.45 or 45%.