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

Key Concepts

Poisson Distribution Formula

The Poisson distribution is a probability model that describes events occurring at a constant average rate independently of the time since the last event. It’s often used to model scenarios in queues, telecommunications, and other systems where events happen sporadically over a period of time.

For a given average rate of occurrence \( \lambda \) and an interval of time, the formula to calculate the probability of exactly \( k \) events occurring is:\[ P(X = k) = \frac{e^{-\lambda t}(\lambda t)^k}{k!} \]Here, \( e \) is the base of the natural logarithm, \( t \) is the time period, \( k \) is the number of events (in the context of this exercise, \( k=0 \) denotes no cars passing), and \( k! \) is the factorial of \( k \) (i.e., the product of all positive integers up to \( k \) ). To handle the scenario of no events occurring, which is our concern here for Al crossing the road safely, we set \( k=0 \) resulting in the formula simplifying significantly as anything to the power of zero is 1, and the factorial of zero is 1.

Converting Units for Probability Calculations

Converting units is a crucial step in many probability calculations, particularly when working with rates that are not in the desired time unit. In our highway crossing example, the rate \( \lambda \) is given in cars per minute, but the times \( s \) are provided in seconds.

To convert the time unit from seconds to minutes, we use the following relation:\[ 1 \text{ minute} = 60 \text{ seconds} \]Therefore, for any given time \( s \) in seconds, the time in minutes \( t \) is calculated by:\[ t = \frac{s}{60} \]This ensures that the units for time in the Poisson distribution formula are consistent with the rate \( \lambda \) provided. Without this step, probabilities calculated would not correspond to the actual scenario, leading to incorrect conclusions.

Probability Calculation Steps

Accuracy in probability calculation is paramount, and this involves a thorough step-by-step approach. Here's an overview of the steps that were used in our example of Al crossing the highway:

• Step 1: Convert units of time - Change the time Al takes to cross the highway from seconds to minutes.
• Step 2: Apply the Poisson distribution formula - With \( \lambda \) representing cars per minute and \( t \) as the converted time in minutes, insert these values into the Poisson distribution formula to find the probability of no cars passing (\( k = 0 \) events).
• Step 3: Simplify and calculate the probability - Since we are interested in the scenario where no car passes by (\( k=0 \) events), the Poisson formula simplifies, letting us easily compute the final probabilities.

Adhering to these steps ensures accurate application of the Poisson distribution, providing students with the ability to evaluate various scenarios systematically.