# Poisson distribution problem

The number of cars driving past a parking area in a one-minute time interval has a Poisson distribution with mean lambda. The probability that any individual driver actually wants to park his or her car is p. Assume that individuals decide whether to park independently of one another.
a)If one parking place is available and it will take you one minute to reach the parking area,what is the probability that a space will still be available when you reach the lot? (Assume that no one leaves the lot during the one-minute interval.)

b)Let W denote the number of drivers who wish to park during a one-minute interval.Derive the probability distribution of W.

could anyone help me to solve this problem? I've no idea about this.

## Answers and Replies

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The number of cars driving past a parking area in a one-minute time interval has a Poisson distribution with mean lambda. The probability that any individual driver actually wants to park his or her car is p. Assume that individuals decide whether to park independently of one another.
a)If one parking place is available and it will take you one minute to reach the parking area,what is the probability that a space will still be available when you reach the lot? (Assume that no one leaves the lot during the one-minute interval.)

b)Let W denote the number of drivers who wish to park during a one-minute interval.Derive the probability distribution of W.

could anyone help me to solve this problem? I've no idea about this.
Try conditioning on N, with

$$P[W=k|N=n] = (^nC_k)p^k(1-p)^{n-k}$$

and P[W=k] is simply the expectation wrt N.