04_probability2

# 04_probability2 - Today Probability II 9.07 The gambler's...

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Probability II 9.07 9/17/2004 Today The gambler’s fallacy Expected value Bayes’ theorem The problem of false positives Making a decision Making a decision the Bayesian way Probability with continuous random variables The gambler’s fallacy • 0.5 | 8 H) = P(T) = 0.5 gambler’s fallacy . believe in probability. the right answer. gambler’s fallacy were true They then switch coins, and flip. flipping), or If you get 8 heads in a row, on a fair coin, what is the probability that the next one will be a tail? Remember, coin tosses are independent P(T But, most people feel like, having tossed 8 heads, it must be more likely you get a tail next, to “even out” the number of heads. This is the Basically, most human beings don’t really, in their gut, But we can learn to rationally get Some paradoxes & puzzles, if the Anna gets 10 heads in a row on her coin. Lee gets 10 tails in a row on his coin. – Is Anna more likely to get a head, to balance out all the tails on Lee’s coin (which she’s now – Is Anna more likely to get a tail, to balance out all the heads she just flipped on her own coin? 1

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Would you carry that coin to the nearest bar, and would come up tails? gambler’s fallacy were true which situation I have: (p(T)>0.5) it up. p(T) = 0.5 Some paradoxes & puzzles, if the If you manage to get 10 heads in a row on a coin, have you “stored up” heads? try to bet someone a lot of money the coin Some paradoxes & puzzles, if the If I get 10 heads in a row on my coin, how do I know The coin was “balanced” before the 10 heads. I expect to see some tails, soon, to balance out the recent heads. Before, the coin had a surplus of tails. The 10 heads I saw evened Now the coin is balanced, and I expect p(T)=0.5. We still have a surplus of tails. I should expect to see more heads, to balance. (p(T)<0.5) In some sense, the fact that you don’t know which situation you’re in means you should consider it equally likely that we now have a surplus of heads, vs. a surplus of tails. Your best guess would be to assume that the coin is now balanced. So, even with the gambler’s fallacy, your best guess should be that The law of averages heads than expected. Don’t we need to toss some tails to get this error down to zero? No, the absolute error actually goes up with more coin tosses. An example: Absolute error = # heads 10 1 10 2 10 3 10 4 -30 -20 -10 0 10 20 30 40 by chance, ½ heads… •I n t h e long run , the relative frequency of an event approximates the probability of that event.
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04_probability2 - Today Probability II 9.07 The gambler's...

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