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Chapter 17 STA 200 Summer I 2011

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Flipping Coins If you toss a coin repeatedly, you expect it to come up heads half the time. Suppose you toss a coin 10 times. – In this scenario, you “expect” to get 5 heads. – Would it be that unusual to get 4 heads or 6 heads instead? What about 3 heads or 7 heads?

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Fundamentals Chance Behavior: – unpredictable in the short run – predictable pattern in the long run Random Phenomenon: – individual outcomes are uncertain – regular distribution of outcomes in a large number of repetitions

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Coin Tossing (cont.) 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 extrapolation of what would happen in an infinitely long series of trials.

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Probability 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 (infinite) 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.

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Myth #1 – Short-Run 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. 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.

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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.

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Law of Averages (cont.) 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. The law of averages doesn’t work for the same reason that short-run 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.

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Law of Large Numbers 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 the number of tosses. This may or may not occur.

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Law of Large Numbers (cont.) For example, the following results are certainly possible: # of tosses# of headsproportion of heads difference between # of heads and half the # of tosses 104.41 10047.473 1,000511.51111 10,0005,074.507474

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Personal/Subjective Probability A personal probability is determined by a person making a judgment regarding the likelihood of an event. The events of interest for personal probability are usually ones that are difficult to repeat (or not repeatable at all). The value of a personal probability will vary from one person to another. Personal probabilities can be found in weather forecasts, stock price projections, and sports predictions.

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