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Unformatted text preview: Chance If you toss a coin repeatedly, you might expect it to come up heads half the time. Suppose you toss a coin 10 times. For the coin to come up heads half the time, you would have 5 heads and 5 tails after 10 tosses. Would it seem strange to get 4 heads instead of 5? How about 6 heads? (For that matter, would 3 heads or 7 heads seem out of the ordinary?) Using a fair coin, the probability of obtaining 5 heads in 10 flips is only about 24.6%. In general, chance behavior is unpredictable in the short run, but it has a regular and predictable pattern in the long run. A phenomenon is random if individual outcomes are uncertain, but there is a regular distribution of outcomes in a large number of repetitions. When repeatedly tossing a coin, the proportion of heads will vary quite a bit at first. After more and more tosses, the proportion will get close to 0.5 and stay there. Probability is an idealization based on an extrapolation of what would happen in an infinitely long series of trials. The probability of any outcome of a random phenomenon is a number between 0 and 1 that describes the proportion of how often the outcome would occur in a long series of repetitions. If an outcome never occurs, it has probability 0. If an outcome always occurs, it has probability 1. Any other type of outcome has a probability somewhere between 0 and 1. Myth #1: ShortRun Regularity The idea of probability is that a random event (like flipping a coin) is regular in the long run. In the short run, this is not the case. For example, if a coin is tossed 10 times, the outcome HTTHTHTTHT looks more probable than HHHHTTTTTT. However, both are equally probable: both outcomes are 4 tails in 10 flips. Myth #1: ShortRun Regularity Even though heads and tails are equally probable, they don't have to come close to alternating in the short run. In other words, the coin has no memory. The probability of the coin coming up heads on any given flip is 0.5, regardless of what has already happened. Myth #2: Law of Averages Usually, when the phrase "law of averages" is used, there is an implication that one or more outcomes have somehow become more likely. For example: If a coin is flipped six times, and comes up heads each time, then by the "law of averages" it's more likely for the coin to come up tails on the seventh flip, in order to compensate for the imbalance between heads and tails. Myth #2: Law of Averages The six consecutive outcomes of heads will never be compensated for; instead, they will be overwhelmed in the long run. If the coin is flipped 1,000 times, the results of the first 6 flips will be overwhelmed by the results of the following 994. Myth #2: Law of Averages Essentially, the law of averages doesn't work for the same reason that shortrun regularity doesn't work: the individual coin flips are independent of each other, so the probability of a coin coming up heads on any given flip is unaffected by what has previously occurred. There is something called the "law of large numbers," which states that the proportion of a specific outcome (like a coin coming up heads) will tend to stabilize as the number of trials increases. However, this isn't the same thing as saying that, for example, the number of heads will tend to get closer to half the number of tosses. This result may or may not occur. For example, the following results are certainly possible:
# of tosses # of heads proportion of heads 10 100 1,000 10,000 4 47 510 5,074 0.4 0.47 0.51 0.5074 difference between # of heads and half the # of tosses 1 3 10 74 So the proportion will tend to stabilize; the count may or may not. ...
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This note was uploaded on 04/17/2008 for the course STA 200 taught by Professor Wierdhobbitguy during the Spring '08 term at Kentucky.
 Spring '08
 wierdhobbitguy

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