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Probability Question?
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?
3 Answers
- 1 decade agoFavorite Answer
Just start breeding the smallest of the mules. Duh!
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The probability of being less than 32 given a mean of 32.5 and a standard deviation of 1.25.
The deviation from mean is -0.5. Divide that by sd for the z-score. -0.5 / 1.25 = -0.4.
The probability of having a z-score less than -0.4...
34.4578%
From a z-score to percentage calculator ( http://www.measuringusability.com/pcalcz.php ), or just look at any z-score table.
The probability of meeting this condition AND the color condition is the product.
34.4578% * 20% =
6.81156% chance of getting a proper height and a proper color out of each mule.
The probability of this not happening, then, is the compliment
93.18844% chance of getting an improper height and/or improper color.
Seven attempts made each year.
93.18844%^7 =
61.0287% chance of the condition NOT being met after seven attempts... that is, improper height and/or improper color seven times in a row.
The compliment of that, is
38.9713% that the condition will be met at least once. Proper height and proper color at least once in seven attempts, in any particular order or quantity.
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Andrew B's reasoning wasnt entirely wrong. The difference between his answer and mine is that he guessed, estimated and approximated each step of the way... using intuition instead of rigorous mathematics. I used proper statistics and probability concepts, with four or five digits of accuracy. He used one digit of accuracy based on intuitive reasoning.
Another difference is that in Andrews answer, he assumed that there would be 3 out of seven mules to meet the height requirement. He based 3 on "slightly less than half" when in fact the deviation in question is "significantly less than half". I would have guessed 2 instead of 3. And if you assume 2 instead, you end up with a result of 40% using Andrews reasoning... which is much more consistent with my answer.
Source(s): HA. Mules are sterile, for those that dont know. - 1 decade ago
32.5 inches with a standard deviation of 1.25 inches = means that a little less than 50% of the mules will be less than 32 inches. With seven foals, assume 3 will be less than 32 inches. Then there is a one in five probabilty of getting the right color. So the probabilty of getting one high priced foal is 3 x 1/5 = 3/5 or 60%.
- asimovLv 61 decade ago
CogitoErgoCogitoSum has 100% right.
if you want a correct answer , use his answer
