A busy restaurant determined that between 6:30 P.M. and 9:00 P.M. on Friday nights, the arrivals of customers are Poisson distributed with an average arrival rate of 4.32 per minute.
What is the probability that at least 1 minute will elapse between arrivals?
Round answer to 4 decimal places.
I know lambda is 4.32 so then do I just plug in 0 for x, get that solution and subtract that from 1.0?
You are correct! Plug in 0 for x and subtract from 1.
Yes, you are correct. To find the probability that at least 1 minute will elapse between arrivals, you can use the Poisson distribution formula. The formula for the probability mass function of the Poisson distribution is:
P(x, λ) = (e^(-λ) * λ^x) / x!
In this case, λ (lambda) represents the average arrival rate of 4.32 per minute.
To find the probability that at least 1 minute will elapse between arrivals, you need to calculate the probability of having 0 arrivals in a given minute. So, you can use x = 0 in the formula.
P(x=0, λ=4.32) = (e^(-4.32) * 4.32^0) / 0!
Since 0! (0 factorial) is equal to 1, the equation becomes:
P(x=0, λ=4.32) = e^(-4.32)
Now, you can calculate this value and subtract it from 1.0 to find the probability that at least 1 minute will elapse between arrivals.
To find the probability that at least 1 minute will elapse between arrivals, you can use the Poisson distribution formula:
P(X = x) = (e^(-lambda) * lambda^x) / x!
In this case, lambda = 4.32 (average arrival rate per minute), and you want to find P(X > 0) since you're interested in the case where at least 1 minute elapses.
To calculate this probability, you need to find P(X = 0) and subtract it from 1.
P(X = 0) = (e^(-4.32) * 4.32^0) / 0!
The exponential function e^(-4.32) can be approximated to 0.013167. And 4.32^0 is equal to 1. Since 0! is equal to 1, the denominator is also 1.
P(X = 0) = 0.013167
Now subtracting this from 1 gives you the probability of at least 1 minute elapsing between arrivals.
P(X > 0) = 1 - P(X = 0)
= 1 - 0.013167
≈ 0.9868
Therefore, the probability that at least 1 minute will elapse between arrivals is approximately 0.9868.