91Ó°ÊÓ

Each of $500$soldiers in an army company independently has a certain disease with probability $1/103$. This disease will show up in a blood test, and to facilitate matters, blood samples from all $500$soldiers are pooled and tested.

Since the sample of soldiers is very large, $n=500$and the probability of the disease for every soldier is relatively low, $p={10}^{-3}$, we can use Poisson approximation. Hence, if we mark $X$as the random variable that marks the number of diseased soldiers, we have that $X$has approximatively $Pois\left(np\right)=Pois(0.5)$distribution.

Thus,

$P(Xâ‰\yen 1)=1-P(X=0)=1-e^{-0.5}≈0.393$.

So the probability that at least one soldier will be tested positive is$0.393$.

Each of $500$ soldiers in an army company independently has a certain disease with probability $1/103$. This disease will show up in a blood test, and to facilitate matters, blood samples from all $500$ soldiers are pooled and tested.

Now, we are given the information that the test is positive, i.e. that $Xâ‰\yen 1$. Now, we have to find the probability of $X>1$given that information.

We have that

$P(X>1âˆ\pounds Xâ‰\yen 1)$

$=\frac{P(X>1,Xâ‰\yen 1)}{P(Xâ‰\yen 1)}$

$=\frac{P(Xâ‰\yen 2)}{P(Xâ‰\yen 1)}$

$=\frac{1-e^{-0.5}-0.5e^{-0.5}}{1-e^{-0.5}}$

$=0.299$.

The probability, under this circumstance, that more than one person has the disease is$0.299$.

Each of$500$ soldiers in an army company independently has a certain disease with probability $1/103$. This disease will show up in a blood test, and to facilitate matters, blood samples from all $500$ soldiers are pooled and tested.

We understand that Jonas carries the disease. But, we accomplish not hold any knowledge concerning any of the other soldiers. Hence, we can exclude Jonas from the story and just assume the remaining $499$ soldiers. But, observe that we want to find the probability that there is more than one soldier with disease and we know that Jonas has the disease, so, we in fact desire to find the probability that some of the remaining $499$ soldiers carry the disease.

So, here repeats story from the part $\left(a\right)$, and since $n$ has been changed just a bit, the required probability also does not change a lot. The answer is $0.393$.

Each of $500$ soldiers in an army company independently has a certain disease with probability $1/103$. This disease will show up in a blood test, and to facilitate matters, blood samples from all $500$ soldiers are pooled and tested.

Do the identical item as in $\left(c\right)$, exclude Jonas and the first $i-1$people from the story. Use $Y$as the random variable that marks the number of a deceased soldier from the remaining $500-i$soldiers. Similarly, $Y~Pois\left((500-i)·{10}^{-3}\right)$.

We want to find the probability that $Yâ‰\yen 1$.

Hence

$P(Yâ‰\yen 1)=1-P(Y=0)=1-e^{-(500-i)·{10}^{-3}}$.

The probability, as a function of$i$is found to be$1-e^{-(500-i)·{10}^{-3}}$.