Problem 2 - A First Course in Probability

Question

Cars cross a certain point in the highway in accordance with a Poisson process with rate \(\lambda=3\) per minute. If Al runs blindly across the highway, what is the probability that he will be uninjured if the amount of time that it takes him to cross the road is \(s\) seconds? (Assume that if he is on the highway when a car passes by, then he will be injured.) Do this exercise for \(s=2,5,10,20\).

Step-by-Step Solution

Answer

The probabilities of Al crossing the road uninjured for each of the times are: - s = 2 seconds: \(0.904837\) - s = 5 seconds: \(0.778801\) - s = 10 seconds: \(0.606531\) - s = 20 seconds: \(0.367879\)

Step 1: Convert the time to minutes

To use the Poisson distribution, we need the time in minutes. Since we have the time in seconds, we will convert it to minutes by dividing by 60.

Step 2: Use the Poisson distribution formula

With the time in minutes, we can find the probability of no cars crossing the highway (k = 0) using the Poisson distribution formula.

Step 3: Calculate the probability

Calculate the probability for each value of s after plugging in the values into the Poisson distribution formula. Let's now calculate the probability for each value of s:

Step 4: Case 1: s=2 seconds

Step 1: Convert the time to minutes: t = 2/60 = 1/30 minutes Step 2: Use the Poisson distribution formula: \(P(X=0) = \frac{e^{-3\times(1/30)}(3\times(1/30))^0}{0!}\) Step 3: Calculate the probability: \(P(X=0) \approx 0.904837\)

Step 5: Case 2: s=5 seconds

Step 1: Convert the time to minutes: t = 5/60 = 1/12 minutes Step 2: Use the Poisson distribution formula: \(P(X=0) = \frac{e^{-3\times(1/12)}(3\times(1/12))^0}{0!}\) Step 3: Calculate the probability: \(P(X=0) \approx 0.778801\)

Step 6: Case 3: s=10 seconds

Step 1: Convert the time to minutes: t = 10/60 = 1/6 minutes Step 2: Use the Poisson distribution formula: \(P(X=0) = \frac{e^{-3\times(1/6)}(3\times(1/6))^0}{0!}\) Step 3: Calculate the probability: \(P(X=0) \approx 0.606531\)

Step 7: Case 4: s=20 seconds

Step 1: Convert the time to minutes: t = 20/60 = 1/3 minutes Step 2: Use the Poisson distribution formula: \(P(X=0) = \frac{e^{-3\times(1/3)}(3\times(1/3))^0}{0!}\) Step 3: Calculate the probability: \(P(X=0) \approx 0.367879\) So, the probabilities of Al crossing the road uninjured for each of the times are: - s = 2 seconds: 0.904837 - s = 5 seconds: 0.778801 - s = 10 seconds: 0.606531 - s = 20 seconds: 0.367879