1. ## Binomial distribution problem

Hi, I have another question, and I just wanted to say thanks again to everyone here who spends their time helping, it really means a lot to me.

The problem:

Blood samples are to be taken from every soldier in a military unit to nd out if any of
the soldiers are infected by a certain disease. The probability that the blood sample from a
randomly chosen soldier is positive (meaning he is infected) is assumed to be p, and the test
results for different individuals are assumed to be independent.

The sampling is done in the following way: Blood samples are taken from k(> 1) soldiers at
a time, the samples are mixed and the mixture is analysed. If the mixture gives a positive
reaction, which means that at least one of the soldiers is infected, new blood samples from
the k soldiers are taken, and the samples are analysed individually. If the mixture gives a
negative reaction, there will not be taken any more samples.

a) Explain why the number of infected soldiers in a k-group is binomially distributed, and
show that the probability of a positive reaction in a mixture of k blood samples is:

1 - (1 - k)p

b) How is the conditional probability of an event A given event B defined? Soldier 19
Haugen is informed that his k-group has shown a positive reaction. What is then the
probability of him being positive?

What I have done to solve it so far:

I know that the number of infected solders in a k-group is binomially distributed because it deals with discrete values, a blood test, in this case, can only be positive or negative. I am, though, not even sure on how to start showing that the probability of a positive reaction is: 1 - (1 - k)p.

For b I know that the conditional probability of an event A given event B is: P(A in union with B) / P(B), but I am not sure how to, from the information I have, know the probability of solider 19 being positive.

2. ## Re: Binomial distribution problem

Hey Nora314.

Hint: A positive reaction means at least 1 soldier has a positive reading. This means that a negative reaction is the complement of this. This means P(Negative Reaction) = P(X < 1) = P(X=0) = 1 - P(X > 0).

3. ## Re: Binomial distribution problem

Ok, let me see,

P(X < 1) = P(X = 0) = 1 - P(X > 0)
P(X > 0) = P(X = 1) = 1 - P(X < 1)
P(X > 0) = 1 - [1 - P(X > 0)]
P(X > 0) = p (from the exercise text)
P(X > 0) = 1 - [1 - p]k (in the power of k because there are k soldiers)

Does this look right? Thank you for the answer by the way!

4. ## Re: Binomial distribution problem

And I think I just miraculously managed b) on my own