In fact, the number of possible outcomes if one were to toss a coin a thousand times is approximately 10.72 300! With a thousand trials, the number of possible outcomes is, of course, much larger. Above, we trialed only twice, so, only four outcomes were possible. What is the likelihood of this outcome - 1,000 coin tosses and 500 show heads while 500 show tails? Look at the number of possible outcomes. This result is exactly the same as if a thousand voters tossed a coin to determine their choice before driving to the polls. Now, extend this analysis, to 1,001 voters - five-hundred votes for Kramer and five-hundred votes (and mine) for Schmuki. Therefore, the probability of the event that I decide the election is 1/4 + 1/4 = 1/2, or. But, two of the events (Kramer-Schmuki and Schmuki-Kramer) would permit my vote alone to determine the outcome. Thus, the probability of any one outcome is 1/4, or. To see this, note that we now have four possible outcomes (Kramer-Kramer, Kramer-Schmuki, Schmuki-Kramer, and Schmuki-Schmuki). The probability of my vote alone electing Schmuki to the Assembly is. One votes for Kramer, while the other votes for Schmuki. First, suppose that only three voters show up on election day. I assume now that the final vote tally (including my vote) is an odd number, in order to avoid a tie. That is, the likelihood of any one event is not contingent upon the results of any other trials. All trials are independent, furthermore, of other trials. Since these events are equally likely, assuming a fair coin, a coin toss over many trials will yield half of the events as heads and half as tails. It can only show either a head or a tail. Why? There are only two possible events resulting from tossing a coin. The probability of tossing a coin and it showing a head or tail is. But the result is the same as if they behaved randomly. Now, in the real world, of course, voters do not behave this way. Note that this is a necessary condition for one voter to decide an election. I ignore the last vote because I assume that that voter (the deciding vote) walks to the polls with a firm mind of his/her choice. Logarithmic FunctionsA tied vote is no different from tossing a coin 14,418 times. It turns out that solving this problem involves multiplying an extremely small number by an equally extreme large number. In order to locate the approximate likelihood of an event like this occurring, we need two pieces of mathematics - one relatively straightforward, the other somewhat obscure. In that case, an individual could decide the election - the outcome would be 7,210 to 7,209. The only outcome contingent upon any one voter deciding between Roger Danielsen or Ann Nischke was 7,209 votes registered for both candidates. Ignoring the Libertarian Party candidate and "scattering" votes, 14,419 citizens here chose either the Democratic or Republican Party candidate. But, why? In 2002, 14,921 voters showed up at the polls to decide who would represent them in the State Assembly for the 97th District. To anyone reading this, have no illusions. With the midterm election fast approaching, I pondered recently why the act of voting is significant at all.
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