SOLUTION: You buy 1000 foam balls and 3 colors of paint from the craft store. Each time you take a new ball out of the bag, it is randomly painted one of your three colors (red, green, blue

Algebra ->  Probability-and-statistics -> SOLUTION: You buy 1000 foam balls and 3 colors of paint from the craft store. Each time you take a new ball out of the bag, it is randomly painted one of your three colors (red, green, blue      Log On


   

Question 698617: You buy 1000 foam balls and 3 colors of paint from the craft store. Each time you take a new ball out of the bag, it is randomly painted one of your three colors (red, green, blue). You group every 3 balls together.
a) How many groups would you expect to have all three balls painted the same color?
b) If your blue can was 75% empty and only had enough paint to cover 200 balls, how many groups would you expect to have 3 red balls?

Answer by Edwin McCravy(20062) About Me  (Show Source):
You can put this solution on YOUR website!
a) How many groups would you expect to have all three balls painted the same color? 
There will be 333%261%2F3 groups.

The expected number of three of the same color is np where n=333%261%2F3 and p is
the probability of choosing 3 balls painted the same color.

Actually to find the probability of choosing 3 we would be choosing without
replacement.  However, since there are 1000 balls, the probability of choosing 
without replacement is approximately the same as choosing with replacement. 

So the probability of selecting 3 reds is

P[(red 1st) AND (red 2nd) AND (red 3rd) = approximately expr%281%2F3%29%2Aexpr%281%2F3%29%2Aexpr%281%2F3%29 = 1%2F27.

and the probabilities of choosing 3 greens and of 3 blues are the same. So the
probability of selecting 3 of the same color is 1%2F27%2B1%2F27%2B1%2F27 = 3%2F27 = 1%2F9.

So the expected number is 333%261%2F3 groups times 1%2F9 or about 37.04

-----------

b) If your blue can was 75% empty and only had enough paint to cover 200 balls, how many groups would you expect to have 3 red balls? 
On the average when the first 600 have been painted, 1%2F3 of them,
or 200 will be red, 200 will be green and 200 will be blue.

But now we're out of blue paint, so on the average, half the 
remaining 400, or 200, will be red and the other 200 green.

So on the average we end up with 400 reds, 400 greens, and 200 blues.

The expected number of three reds is np where and n = 333%261%2F3
and p is the probability of selecting 3 reds.

Again since there are 1000 balls, the probability of choosing without 
replacement is approximately the same as choosing with replacement.

The probability of selecting a red ball is 400%2F1000 = 2%2F5

P[(red 1st) AND (red 2nd) AND (red 3rd)] = approximately expr%282%2F5%29%2Aexpr%282%2F5%29%2Aexpr%282%2F5%29 = 8%2F125.

So the expected number is 333%261%2F3 groups times 8%2F125 
or about 21.33.
 
Edwin