Probability of Producing High-Priced Miniature Mules from Breeding Stock

[mossfan563]

Homework Statement

You breed miniature mules (father is a miniature donkey, mother is a miniature horse) and currently have as breeding stock 1 jack (male donkey) and 7 mares (female horses) and these are able to produce 7 mule foals (baby mules) for you every year. You would like to produce miniature mules that are quite small, but unfortunately there is considerable variation in the sizes of mule offspring produced even from the same set of parents. Based on the best information you can obtain, the heights of the mules you produce are normally distributed with a mean height of 32.5 inches and a standard deviation of 1.25 inches. Any mule less than or equal to 32 inches can be sold for a high price and any mule greater than 32 inches in height can be sold only for a lower price.

If a mule foal is both less than 32 inches in height and of a desirable color it can be sold for a very high price. You have determined that there is a 20% probability that any given foal will be of a desirable color, and that color is independent of height. What is the probability that you will have at least one mule that can be sold for a very high price in any given year?

Homework Equations

phi((x - avg)/(std dev.))
phi((x + .5 - np)/(sqrt(np(1 - p))))

The Attempt at a Solution

I know the probability of any given mule being sold at a high price when it's less than 32 inches. I just don't know what to do with the other probability that's given in the question. What's the correct formula to use or start with? I just need a point in the right direction.

 

[PhaseShifter]

Break it down into parts:

1) What is the probability of a given mule being under 32 inches?
2) What is the probability of a given mule being a desirable color? [given]
3) Given 1 and 2, what is the probability that a given mule is both a desirable color and less than 32 inches [hint: what does "color is independent of height" mean in terms of combined probability?]
4) What is the probability that a given mule does not meet the criteria for question 3? [hint: p(not a)=1-p(a)]
5) What is the probability that all 7 mules born in one year do not meet the criteria for question 3? [assume each foal is independent of the others]
6) What is the probability that 5) turns out not to be the case? [see the hint for 4]