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