Probability of Sky Falling is 0/0?
Date: 3 Mar 1995 19:46:02 -0500 From: Anonymous Subject: The sky will fall Dear Dr. Math, We teach an 8th grade math class studying probability. We gave a homework assignment asking students to assign probabilities to various situations. When asked the probability that the "sky will fall" one student responded "0/0." He was convinced that his answer was correct since he believed that there was NO possibility of the sky falling. What should we have told this 8th grade student about dividing by zero? And is 0/0 different than 1/0? Is there a way to address this in an 8th grade math class drawing on the knowledge that they have?
Date: 5 Mar 1995 13:39:53 -0500 From: Dr. Ken Subject: Re: The sky will fall Hello there! Here are my thoughts about that interpretation of probabilities. To find the probability of something happening, you find out how many different ways the thing you're looking at can happen, and then divide that by the total number of things that can happen. So you'd have zero, the number of different ways the sky can fall (if you really think it can't fall), divided by however many things you think CAN happen, which is probably some positive number. In any case, the denominator can't be zero, since that means that there are zero things that can happen. The difference between 0/0 and 1/0 is sometimes complicated. For instance, if I look at these two sequences: Sequence A: 5*1 5*1/2 5*1/3 5*1/4 5*1/5 ----,------,------,------, ------, ... 1 1/2 1/3 1/4 1/5 Sequence B: 1 1 1 1 1 ---, -----, -----, -----, -----, ... 1 1/2 1/3 1/4 1/5 The numerator and the denominator in Sequence A are both going to zero. So this sequence should be getting closer and closer to 0/0. But notice that every term in this sequence is 5. So the limit of this sequence is 5. And I could have replace the 5 with any number I want, to get whatever limit I want for 0/0. So if you say that the probability of something happening is 0/0, you might be saying that it has probability 1, or .5, or anything at all. In the second sequence though, notice that we could write it as 1,2,3,4,...which goes to infinity. The only way we can interpret 1/0 (or any nonzero real number over zero) is as positive or negative infinity. In general though, listen to the oft-heard words of advice: DON'T DIVIDE BY ZERO, but with the following addendum: if you do, make sure you're prepared to deal with what happens. In your case, you could have caused the sky to fall, because you said that the probability of it happening might be 1. -Ken "Dr." Math
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