# a little harder

• Oct 29th 2007, 05:34 AM
slevvio
a little harder
Thanks again to those that responded to my other posts:) I am having trouble with this question however.

Small particles can be found in the molten glass from which glass bottles are made. If even one of these particles is incorporated into a bottle, then the bottle must be scrapped. Suppose that ten bottles are to be produced from a quantity of molten glass in which three of these particles are randomly distributed. What is the expected number of bottles that will have to be scrapped? (Let the random variable X be the number of bottles in which there is at least one particle).

I know how to work out the expectation, but I have no idea how to work out the probability distribution for x=1,2,3. Thanks a lot any help would be appreciated.
• Oct 29th 2007, 11:02 PM
CaptainBlack
Quote:

Originally Posted by slevvio
Thanks again to those that responded to my other posts:) I am having trouble with this question however.

Small particles can be found in the molten glass from which glass bottles are made. If even one of these particles is incorporated into a bottle, then the bottle must be scrapped. Suppose that ten bottles are to be produced from a quantity of molten glass in which three of these particles are randomly distributed. What is the expected number of bottles that will have to be scrapped? (Let the random variable X be the number of bottles in which there is at least one particle).

I know how to work out the expectation, but I have no idea how to work out the probability distribution for x=1,2,3. Thanks a lot any help would be appreciated.

Number the particles. Then let particle 1 be in bottle n, then the probability
that all three are in the one bottel is (1/10)x(1/10) that is the probability
that the other two which are equally likley to be in any bottle are both also in bottel n.

Thus the probability that they are all in the same bottle is 0.01, which is the
probability that x=1.

The probability that 2 is in bottle n and that 3 is not is (1/10)x(9/10) as
is the probability that 3 is in bottle n and 2 is not, so the probability that
there are two particles in one bottle and one in another is 18/100.

Thus the probability that they are two in one bottle and one in another is
0.18, which is the probability that x=2.

The probability that p(x=3)=1-p(x=1)-p(x=2).

RonL