for a small regional airline, the number of claims for missing luggage averages 70 per week.
a. what is the probability that on a given day, there are fewer than 3 claims made against the airline?
b. what is the probability that there are more than 3 claims against the airline on a given day?
Could you please explain the formula required to work this out I am having difficulty understanding.
Much appreciated Jen
Certainly, Jen!
To solve these probability problems, we will use the Poisson distribution. The Poisson distribution is used to model the number of events that occur in a fixed interval of time or space, given the average rate of occurrence.
In this case, we are given that the average number of claims for missing luggage for the small regional airline is 70 per week. Since we want to calculate the probability for a single day, we need to convert this average rate to a daily rate.
To convert from weekly to daily rate, we divide the average weekly rate by 7 (since there are 7 days in a week):
Average daily rate = 70 / 7 = 10 claims per day
Using this average daily rate, we can proceed to answer the two questions:
a. What is the probability that on a given day, there are fewer than 3 claims made against the airline?
To calculate this probability, we will use the Poisson distribution formula:
P(x; λ) = (e^(-λ) * λ^x) / x!
where:
- P(x; λ) is the probability of x events occurring
- e is the base of the natural logarithm (approximately 2.71828)
- λ (lambda) is the average rate of events occurring in the given time interval
- x is the number of events we are interested in
In this case, we want to find P(x < 3), which means the probability that there are fewer than 3 claims made against the airline on a given day. We will calculate this as the sum of P(x = 0), P(x = 1), and P(x = 2).
P(x < 3) = P(x = 0) + P(x = 1) + P(x = 2)
= (e^(-λ) * λ^0) / 0! + (e^(-λ) * λ^1) / 1! + (e^(-λ) * λ^2) / 2!
Now substituting the value of λ (10 claims per day) into the formula, we can calculate each term and sum them up to find the answer.
b. What is the probability that there are more than 3 claims against the airline on a given day?
To calculate this probability, we need to find P(x > 3), which means the probability that there are more than 3 claims made against the airline on a given day. We can calculate this probability as 1 - (P(x ≤ 3)).
P(x > 3) = 1 - P(x ≤ 3)
To obtain P(x ≤ 3), we calculate the sum of P(x = 0), P(x = 1), P(x = 2), and P(x = 3). Then subtracting this sum from 1 will give us the desired probability.
I hope this explanation clarifies the formula and process for you. If you have any further questions, feel free to ask!