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QUESTION IMAGE

the graph to the right is the uniform probability density function for …

Question

the graph to the right is the uniform probability density function for a friend who is x minutes late.
(a) find the probability that the friend is between 25 and 30 minutes late.
(b) there is a 50% probability the friend will arrive within how many minutes?
(a) this probability that the friend is between 25 and 30 minutes late is
(type an integer or a decimal. round to three decimal places as needed.)
(b) there is a 50% probability the friend will arrive within
minutes.
(type a whole number.)

Explanation:

Step1: Recall the formula for uniform distribution probability

For a uniform distribution \(U(a,b)\), the probability density function \(f(x)=\frac{1}{b - a}\), and the probability \(P(c<X<d)=\frac{d - c}{b - a}\).
Assume the uniform distribution is \(U(0,60)\) (since time can range from \(0\) to \(60\) minutes in a general - time - interval context).

Step2: Calculate the probability for part (a)

Here, \(a = 0\), \(b=60\), \(c = 25\), \(d = 30\).

$$P(25

Step3: Calculate the time for part (b)

We know that \(P(0<X<t)=0.5\). Using the formula \(P(0<X<t)=\frac{t - 0}{60-0}\).
Set \(\frac{t}{60}=0.5\). Solving for \(t\), we get \(t = 30\)

Answer:

(a) \(0.083\)
(b) \(30\)