**Exercise 2.34: **We roll a standard fair dice over and over. What is the expected number of rolls until the first pair of consecutive sixes appears? (*Hint*: The answer is not 36.)

**Solution: **Let be the expected number of rolls until the first pair of consecutive sixes appears, the random variable takes on value if the th roll is a and otherwise.

becomes

**Exercise 2.25: **A blood test is being performed on individuals. Each person can tested separately, but this is expensive. Pooling can decrease the cost. The blood samples of people can be pooled and analyzed together. If the test is negative, this one test suffices for the group of individuals. If the test is positive, then each of the person must be test separately and thus total tests are required for the people.

Suppose that we create disjoint groups of people(where divides ) and use the pooling method. Assume that each person has a positive result on the test independently with probability .

**(a)** What is the probability that the test for a pooled sample of people will be positive.

**(b) **What is the expected number of tests necessary?

**(c) **Describe how to find the best value of .

**(d)** Give an inequality that shows for what values of pooling is better than just testing every individual.

**Solution:**

**(a) ** is the probability that the test for a pooled sample of will be negative, so is the probability that the test will be positive.

**(b)** Let . Let be random variable that take on if the test for the th group is positive and if the test is negative. Let , be the total number of tests needed.

It easy to see that .

is a binomial random variable with parameter and , thus .

This yields

.

**(c)*** * We will find among divisors of the value that minimizes .

If is that value then we have and .

This leads to .

This leads to .

Then we obtain .

This leads to

If we have value of then for all divisors of that satisfy , we calculate

and then choose corrersponding to the smallest .

For example, if then . But if then (too large).

Anyone who knows how to solve this problem please help me.

*/* Wrong
*

**(d) **The total number of tests needed if we test every individual is . Assume that we choose to be the greatest divisor of , so that we don’t need to worry about .

We have .

Pooling is better than just testing every individual if

.

*/

Posted by muie geoana on December 12, 2009 at 11:04 am

Part c is wrong.

You are differentiating by k, so d/dx ((1-p)^k) = (1-p)^k ln(1-p), not k (1-p)^(k-1) as you said.

Posted by muie geoana on December 12, 2009 at 11:06 am

replace d/dx by d/dk in the previous post

Posted by thethong on December 12, 2009 at 2:12 pm

Thank you!

I will try to fix it.