Questions: The duration of a professor's class has continuous uniform distribution between 19.2 minutes and 55.5 minutes. If one class is randomly selected and the probability that the duration of the class is longer than a certain number of minutes is 0.506, then find the duration of the randomly selected class. If P(x>c)=0.506, then find c, where c is the duration of the randomly selected class. Round your answer to one decimal place. c= × minutes

Solution Steps

The duration of the professor's class follows a continuous uniform distribution defined on the interval \( [a, b] \), where \( a = 19.2 \) minutes and \( b = 55.5 \) minutes.

We need to find the value of \( c \) such that the probability \( P(x > c) = 0.506 \). The probability of a random variable \( x \) being greater than \( c \) in a uniform distribution can be expressed as:

\[ P(x > c) = \frac{b - c}{b - a} \]

Setting the equation equal to the given probability:

\[ \frac{b - c}{b - a} = 0.506 \]

Substituting the values of \( a \) and \( b \):

\[ \frac{55.5 - c}{55.5 - 19.2} = 0.506 \]

Calculating \( b - a \):

\[ b - a = 55.5 - 19.2 = 36.3 \]

Now, substituting this back into the equation:

\[ \frac{55.5 - c}{36.3} = 0.506 \]

Multiplying both sides by \( 36.3 \):

\[ 55.5 - c = 0.506 \times 36.3 \]

Calculating \( 0.506 \times 36.3 \):

\[ 0.506 \times 36.3 \approx 18.3738 \]

Thus, we have:

\[ 55.5 - c = 18.3738 \]

Rearranging to solve for \( c \):

\[ c = 55.5 - 18.3738 \approx 37.1262 \]

Rounding \( c \) to one decimal place gives:

\[ c \approx 37.1 \]

Final Answer