Poisson Probability: At Least 50% Defective Brake Lights

[Amannequin]

Homework Statement

Suppose that 1% of cars have defective brake lights and n cars are to be inspected. How large should n be for the sample to have a probability of at least 50% of containing a car with a defective brake light? Give an answer using a Poisson approximation with an appropriate mean.

The Attempt at a Solution

Let X-Bin(n, 0.01).
We can approximate X with the Poisson distribution assuming n large and with mean 0.01n.
That is, X≈Po(0.01n).
We want P(X=1)≥ 0.5 which yields ne^-0.01n ≥ 50.

Then I'm stuck. Is this correct so far and any direction on where to go from here will be appreciated. Thanks.

 

[pasmith]

Homework Statement

Suppose that 1% of cars have defective brake lights and n cars are to be inspected. How large should n be for the sample to have a probability of at least 50% of containing a car with a defective brake light? Give an answer using a Poisson approximation with an appropriate mean.

The Attempt at a Solution

Let X-Bin(n, 0.01).
We can approximate X with the Poisson distribution assuming n large and with mean 0.01n.
That is, X≈Po(0.01n).
We want P(X=1)≥ 0.5 which yields ne^-0.01n ≥ 50.

Then I'm stuck. Is this correct so far and any direction on where to go from here will be appreciated. Thanks.

I think you actually want P(X \geq 1) \geq 0.5, since a sample which contains more than one defective car contains a defective car.