Probability of Producing High-Priced Miniature Mules from Breeding Stock

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In summary, the probability of having at least one mule that can be sold for a very high price in any given year can be calculated by breaking down the problem into smaller parts: determining the probability of a mule being under 32 inches, the probability of a mule being a desirable color, and the combined probability of a mule being both under 32 inches and a desirable color. It is important to note that the color of a mule is independent of its height, meaning that it does not affect the probability of a mule being under 32 inches. By using the formula for calculating combined probabilities and the concept of independent events, the probability of having at least one mule that meets the criteria can be determined.
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mossfan563
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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.
 
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  • #2
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]
 

1. What is the "Mule Probability Question"?

The "Mule Probability Question" is a hypothetical scenario used in probability and statistics to illustrate the concept of independent events and the multiplication rule.

2. How does the "Mule Probability Question" work?

The question asks what the probability is of a female mule giving birth to a male offspring, given that mules are known to be infertile. This question demonstrates how the probability of a series of events occurring together is the product of their individual probabilities.

3. What is the importance of the "Mule Probability Question" in understanding probability?

The "Mule Probability Question" helps to illustrate the concept of independence in probability calculations. It also highlights the importance of considering all possible outcomes and their individual probabilities when calculating the probability of a series of events occurring together.

4. Can the "Mule Probability Question" be applied to real-life situations?

While the question itself is hypothetical, the concept of the multiplication rule and independence can be applied to various real-life scenarios, such as predicting the probability of a specific genetic trait being passed down in a family.

5. Are there any variations of the "Mule Probability Question"?

Yes, there are many variations of the "Mule Probability Question" that can be used to demonstrate the same concept, such as flipping multiple coins or drawing cards from a deck without replacement.

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