You are driving a car and the speed is an exponentialy distributed random variable
with mean 50 (in miles per hour). The speed limit is 60 miles per hour, and if your speed is
x > 60 miles per hour, the probability you are stopped by the police is 1 - e^(6-(10/x)), and if you
are indeed stopped you will pay the fine of 100(x - 60) dollars. Find the probability you are
stopped and pay more than 1000 dollars.
To find the probability of being stopped and paying more than $1000, we need to find the probability of two events happening:
1. The event of being stopped by the police
2. The event of paying more than $1000
Let's break down the problem step by step:
Step 1: Find the probability of being stopped by the police.
The probability of being stopped by the police is given by the equation: P(stop) = 1 - e^(6 - (10/x))
Step 2: Find the probability of paying more than $1000 if stopped.
If you are stopped, you will pay a fine of 100(x - 60) dollars. So, we need to find the probability that the fine is greater than $1000. In other words, we need to find P(fine > 1000) = P(100(x - 60) > 1000).
Step 3: Combine both probabilities.
To find the probability of two independent events happening, we multiply the probabilities of each event. So, the probability of being stopped and paying more than $1000 is P(stop) * P(fine > 1000).
Now, let's solve these steps and calculate the result:
Step 1: Find the probability of being stopped by the police:
P(stop) = 1 - e^(6 - (10/x))
Step 2: Find the probability of paying more than $1000 if stopped:
P(fine > 1000) = P(100(x - 60) > 1000)
To solve this equation, we can divide both sides by 100:
x - 60 > 10
x > 70
So, if you are stopped, you will pay more than $1000 when your speed is greater than 70 miles per hour.
Step 3: Calculate the combined probability:
P(stop and fine > 1000) = P(stop) * P(fine > 1000)
P(stop and fine > 1000) = [1 - e^(6 - (10/x))] * [1 - P(x ≤ 70)]
To calculate this combined probability, we need more information about the distribution of the random variable "speed" such as the distribution function or the range of speeds considered.
Please provide additional information about the distribution of the speed variable, and I can help you calculate the final probability.