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On the average, only 1 person in 1000 has a particular rare blood type.

(a) Find the probability that, in a city of 10,000 people, no one has this blood type.

The probability can be calculated as \(P=0.999^{10000}=0.0000451733\)

So the probability that no one in the city has this blood type is approximately 0.0045%.

(b) How many people would have to be tested to give a probability greater than 1/2 of finding at least one person with this blood type?

The probability of finding at least one person with the blood type is 1 minus the probability that nobody has it. The probability that nobody has it in a sample of \(n\) people is \(0.999^n\). Therefore, we want to find \(n\) such that:

\(1-0.999^n>0.5 \rightarrow 0.5>0.999^n \rightarrow 0.999^n<0.5\)

We can determine this by solving the equation \(0.999^n=0.5\) using \(log_{0.999}(0.5)\). This results in 692.8005, therefore we would need to test 693 people to have a probability greater than 1/2 of finding at least one person with this blood type.

Testing this solution:

The probability of finding at least one person with the blood type after testing 692 people is \(1-0.999^{692}=0.4995994\)

The probability of finding at least one person with the blood type after testing 693 people is \(1-0.999^{693}=0.5000998\)

This confirms the solution.