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Problem 14

On February 2nd, groundhogs all across the country awake from a long winter sleep to look for their shadows. If a groundhog in a given area sees his shadow, he decides that another six weeks of winter must be on its way, and he returns to his hole. If the groundhog does not see his shadow, he concludes that spring must be near, so he stays above ground.

Looking at the statistics from 2003, it was found that when a longer winter occurred in a given area, the groundhog in that area had a 99 percent chance of seeing his shadow (wow! those groundhogs make great meteorologists). However, in areas that had shorter winters, groundhogs still had a 17 percent chance of seeing their shadows which led to incorrect predictions about the length of the winter ahead. It is also known that 25 percent of the areas tested by groundhogs last year had longer winters. So, if a groundhog in a certain area saw his shadow last year and retreated to his hole, what is the probability that a long winter was actually on its way?

Solution

Note that four event combinations are possible: (1) the groundhog sees his shadow and a long winter occurs, (2) the groundhog sees his shadow and a short winter occurs, (3) the groundhog doesn't see his shadow and a long winter occurs, and (4) the groundhog doesn't see his shadow and a short winter occurs. We know that the groundhog saw his shadow, so only (1) or (2) could have happened. In our sample space, the probability of (1) is (.25)(.99)=0.2475 and the probability of (2) is (.75)(.17)=0.1275. We are interested in when (1) happened given that (1) or (2) surely happened which is 0.2475/(0.2475+0.1275)= .66, so there is a 66 percent chance a long winter was on its way. To understand why this works, look at the special case where we have a sample space of 10000 areas tested by groundhogs. Then 2500 areas had long winters while 7500 had short winters. Then the groundhog saw his shadow in 99 percent of the 2500 long winter areas (that is 2475 of them) and 17 percent of the 7500 short winter areas (that is 1275 of them). We are told that a certain groundhog saw his shadow, so he must be one of the 3750=2475+1275. So the probability that he was one of the 2475 who saw a shadow and experienced a long winter is 2475/3750 = .66 . For more information on this type of problem, do an internet search for Bayes' Theorem.