Probability- Gravner- 80.b

Question

After your complaint about their service, a representative of an insurance company promised to call you "between $7$ and $9$ this evening." Assume that this means that the time $T$ of the call is uniformly distributed in the specified interval

(b) At $8.30$, the call still hasn't arrived. What is the probability that it arrives in the next  $10$  minutes?

Answer 1

Imagine a line with 120 time units and probability of marking a dot on it is uniformly distributed over that line.

Dot is the call by the representative and 120 are the minutes between $7$ PM and $9$ PM.

Now it is given in the question that at $8.30$ PM call is yet to arrive. So by this information we can concentrate all the probability of call arrival between 90 - 120 time units. $(8.30 - 9.00)$

$P(x<100 | x \geq90) =$$\large \frac{10}{30} = \frac{1}{3}$

Answer 2

The PDF of a uniform distribution between 7 PM and 9 PM is:

f(t) = 1/2, for 7 <= t <= 9 & 0, otherwise

We want to find the probability of the call arriving between 8:30 PM and 8:40 PM, given that it hasn't arrived yet. This is a conditional probability.

Conditional Probability:

P(A|B) = P(A and B) / P(B)

In our case:

• A: The call arrives between 8:30 PM and 8:40 PM.
• B: The call hasn't arrived by 8:30 PM.

Calculating P(A and B):

This is the probability that the call arrives between 8:30 PM and 8:40 PM. We can calculate this by integrating the PDF over this interval:

P(A and B) = ∫[8.5, 8.667] (1/2) dt = (1/2) * (8.667 - 8.5) = 0.0833

Calculating P(B):

This is the probability that the call hasn't arrived by 8:30 PM. Since the call is equally likely to arrive at any time between 7 PM and 9 PM, and we know it hasn't arrived by 8:30 PM, the probability that it hasn't arrived is the ratio of the remaining time interval (30 minutes) to the total time interval (120 minutes):

P(B) = 30/120 = 0.25

Calculating the Conditional Probability:

P(A|B) = P(A and B) / P(B) = 0.0833 / 0.25 = 0.3333

Therefore, the probability that the call arrives in the next 10 minutes, given that it hasn't arrived yet, is approximately 0.3333 or 33.33%.