Answer :
The probability, P(X < 51.7) that the class length is less than 51.7 minutes is 0.85
We have that
The distribution is continuous uniform between 50.0 and 52.0 minutes
This means that the probability density function (pdf) is constant over this interval.
The formula for the probability density function of a continuous uniform distribution is represented as
[tex]\[ f(x) = \dfrac{1}{b - a} \][/tex]
Where a = 50.0 and b = 52.0
So, we have
[tex]\(f(x) = \dfrac{1}{52.0 - 50.0}[/tex]
[tex]\(f(x) = \dfrac{1}{2}[/tex] for [tex]\(50.0 \leq x \leq 52.0\).[/tex]
The required probability is then calculated as
[tex]\[ P(X < 51.7) = \int_{50.0}^{51.7} f(x) \, dx \][/tex]
So, we have
[tex]\[ P(X < 51.7) = \int_{50.0}^{51.7} \frac{1}{2} \, dx \][/tex]
Integrate
[tex]\[ P(X < 51.7) = \frac{1}{2} \left[ x \right]_{50.0}^{51.7} \][/tex]
Expand
[tex]\[ P(X < 51.7) = \frac{1}{2} \left[ 51.7 - 50.0 \right] \][/tex]
This gives
[tex]\[ P(X < 51.7) = \frac{1}{2} (1.7) \][/tex]
Evaluate
[tex]\[ P(X < 51.7) = 0.85 \][/tex]
Hence, the probability that the class length is less than 51.7 minutes is 0.85
