Here are some interesting mathematical facts:

1) The term "undefined" is distinct from "indeterminate." For example, 1/0 is considered undefined while 0^0 is considered indeterminate, just to name a few examples. Without getting technical (this would involve notions of limits of functions), we can argue that 1/0 is not defined because intuitively, we want to be able to multiply both sides of the equation 1/0=x by 0 to derive some statement. However, when we do so, we get 1/0*0=x*0=0; in other words, we get 1=0. This is obviously illogical. In fact, this will happen whenever you have a nonzero numerator and a zero denominator. There is no number that when multiplied by 0 will yield a nonzero value at least in our conventional system. Thus, 1/0, 2/0 and other such forms are not DEFINED---hence we use the term "undefined." Note that if the numerator and denominator are 0, the situation is very different. That is if we have 0/0=x, and we multiply both sides by 0 to obtain 0/0*0=x*0=0, we get the true statement 0=0; but this works for ANY x. That is to say we want to assign some value to 0/0 such that, intuitively, this value multiplied by 0 (the denominator) yields 0 (the numerator); however, such a value must necessarily be ANYTHING. We cannot DETERMINE the value so we use the term "indeterminate" in such a situation. You can learn more about it @ http://www.khanacademy.org/math/trigonometry/functions_and_graphs/undefined_indeterminate/v/undefined-and-indeterminate.

2) A good probability question for beginners is: Given that an unbiased coin is flipped 10 times, and the coin lands on heads the first 9 times, how likely is the coin to land on heads relative to tails on the tenth time? We might initially expect it to be very unlikely to land on heads relative to tails since the coin has already landed on heads "enough." However, the chance of obtaining heads on the tenth trial is the SAME as that of obtaining tails since each trial is INDEPENDENT of one another. That is to say, even if the coin lands on heads coincidentally on 9 trials in a row, because the coin is unbiased and each trial is independent of the previous one, the tenth trial makes no difference. The probability of landing on heads on the tenth trial is equal to that of obtaining tails which is equal to 1/2.

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