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QuestionFind the expression for the probability of 240 cars parked in Lot A, given the mean is 200. Options include: a. e240/200\mathrm{e}^{240} / 200 ! b. 240!/200240 ! / 200 ! c. e−200200240/240\mathrm{e}^{-200} 200^{240} / 240 ! d. e−240240200/200e^{-240} 240^{200} / 200 !

Studdy Solution

STEP 1

Assumptions1. The mean number of cars parked in Parking Lot A is200. We are asked to find the probability of having240 cars parked in a day3. The number of cars parked follows a Poisson distribution, which is a reasonable assumption for this type of problem

STEP 2

The Poisson distribution is given by the formula(k;λ)=λke−λk!(k; \lambda) = \frac{\lambda^k e^{-\lambda}}{k!}where- kk is the actual number of events (in this case, the number of cars parked) - λ\lambda is the average rate of occurrence (the mean number of cars parked) - ee is the base of the natural logarithm (approximately equal to2.71828) - k!k! is the factorial of kk

STEP 3

We need to find the probability of having240 cars parked, given a mean of200. So, we need to substitute k=240k =240 and λ=200\lambda =200 into the Poisson distribution formula.
(240;200)=200240e−200240!(240;200) = \frac{200^{240} e^{-200}}{240!}

STEP 4

Looking at the options given, we can see that option c. matches the expression we derivede−200200240/240!\mathrm{e}^{-200}200^{240} /240 !So, the correct answer is option c.

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