As in an earlier exercise, busy people arrive at the park according to a Poisson process with rate /hour. Relaxed people arrive at the park according to an independent Poisson process with rate /hour. Assume that no other people arrive at the park.
During the last 10 minutes, exactly two people arrived at the park. What is the probability that they are both relaxed?
4/25 Or 0.16
To find the probability that both people who arrived at the park are relaxed, we can use Bayes' theorem.
Let's denote the event that two people arrive at the park in the last 10 minutes as A, and the event that both people are relaxed as B.
We need to find P(B|A), which is the probability that both people are relaxed given that two people arrived at the park.
According to Bayes' theorem:
P(B|A) = (P(A|B) * P(B)) / P(A)
P(A|B) represents the probability of two people arriving at the park in the last 10 minutes given that they are both relaxed.
P(B) represents the probability that both people are relaxed.
P(A) represents the probability of two people arriving at the park in the last 10 minutes.
Given that the arrivals of busy people and relaxed people are independent and follow Poisson processes, the number of arrivals in a given time period follows a Poisson distribution.
Let's assume the rate of arrival for busy people is λ1 and the rate of arrival for relaxed people is λ2.
P(A|B) is the probability of having two arrivals in the last 10 minutes given that they are both relaxed. This can be calculated as follows:
P(A|B) = (λ2 * 10/60)^2 * e^(-λ2 * 10/60) / (2!)
P(B) is the probability that both people are relaxed. This can be calculated as:
P(B) = (λ2 * 10/60)^2 * e^(-λ2 * 10/60) / (2!)
P(A) is the probability of having two arrivals in the last 10 minutes, which can be calculated as:
P(A) = (λ1 * 10/60 + λ2 * 10/60)^2 * e^(-(λ1 * 10/60 + λ2 * 10/60)) / (2!)
Finally, we can substitute these values into Bayes' theorem to find P(B|A).
To find the probability that both of the two people who arrived at the park are relaxed, we need to use the concept of conditional probability. Let's break down the steps to solve this problem:
Step 1: Understand the problem.
Based on the information given, busy people arrive at the park according to a Poisson process with a rate of λ_busyness (per hour), and relaxed people arrive at the park according to an independent Poisson process with a rate of λ_relaxed (per hour). We're asked to find the probability that the two people who arrived in the last 10 minutes are both relaxed.
Step 2: Identify the relevant Poisson distribution.
The Poisson distribution is defined by the formula P(x; λ) = (e^(-λ) * λ^x) / x!, where P(x; λ) is the probability of getting x events in a given time interval, and λ is the average rate of events per unit time.
Step 3: Calculate the expected number of arrivals in the last 10 minutes.
Since the average rate is given per hour, we need to convert it to the rate per 10 minutes. We can do this by dividing λ by 6, as there are 6 ten-minute intervals in an hour. Let's denote this as λ_relaxed_10min for relaxed people.
Step 4: Calculate the probability of two arrivals in the last 10 minutes.
We'll use the Poisson distribution formula to calculate the probability of two arrivals in the last 10 minutes for both relaxed and busy people. Let's denote these probabilities as P_relaxed_2 and P_busy_2, respectively.
P_relaxed_2 = (e^(-λ_relaxed_10min) * λ_relaxed_10min^2) / 2! = (e^(-λ_relaxed_10min) * λ_relaxed_10min^2) / 2
P_busy_2 = (e^(-λ_busy_10min) * λ_busy_10min^2) / 2! = (e^(-λ_busy_10min) * λ_busy_10min^2) / 2
Step 5: Calculate the probability that both arrivals are relaxed.
The probability that both arrivals are relaxed can be calculated using conditional probability. Given that there were exactly two arrivals in the last 10 minutes, the probability that both are relaxed is determined by the ratio of P_relaxed_2 to the total probability of two arrivals, which is the sum of P_relaxed_2 and P_busy_2.
Probability_both_relaxed = P_relaxed_2 / (P_relaxed_2 + P_busy_2)
By following these steps, you can calculate the probability that both people who arrived in the last 10 minutes are relaxed.