#1
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A question from prob/stat final
A 'classic' prob stat question....
A bag holds three coins. One is double-headed and the other two are fair. A coin is selected at random and flipped four times. What is the probability that the coin is double-headed if it is observed that the coin lands heads up on each of the four tosses? This is from last Spring's final. See if you can get it. (http://www.math.gatech.edu/~belinfan.../pdf/final.pdf) |
#2
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Re: A question from prob/stat final
This is pretty simple if you think about what could have happened. You can draw a coin in any of three ways, and the four flips could come up any of 16 ways. That's 48 possible outcomes. 16 of them involve picking the double headed coin, 2 are picking one of the other two coins and getting four heads by chance. Therefore, 16 of the 18 possibilities mean you picked the double-headed coin, so the posterior probability is 16/18 or 0.8888.
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#3
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Re: A question from prob/stat final
Let A be the event "Double headed coin selected" and B be the event "Four heads thrown" ...
Then the formula P(A|B) = P(B|A) P(A)/P(B) reduces to P(A|B) = P(A)/P(B) {1} Since P(A) = 1/3 and P(B) = P(B|A) P(A) + P(B|not A) P(not A) = 1 * 1/3 + 1/16 * 2/3 = 18/48 Substituting into {1} gives P(A|B) (1/3)/(18/48) = 16/18 |
#4
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A better question would be . . .
If Aaron answers this question, what is the probablity he got it right ?
Based on past history I'd put it at 99%. * A few others . . . Chance that he phrased the answer in a clear and concise fashion ? 99.9% Chance that he got pissy and insulted either the OP or any of the subsequent respondents ? < 0% Chance that I'll read his answer before almost any other ? 100% * - I hear he's also kind to dumb animals. Oh, wait, that's redundant; I already said he never gets pissy toward anyone here. [img]/images/graemlins/smirk.gif[/img] |
#5
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Re: A question from prob/stat final
It depends whether this occured live or online since we all know
ONLINE COIN TOSSING IS RIGGED ! [img]/images/graemlins/mad.gif[/img] |
#6
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Re: A better question would be . . .
Thanks for the kind words. I'm tempted to demonstrate the 1%, just to avoid getting a reputation as a wimp.
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#7
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Re: A question from prob/stat final
So, you have THREE coins and you pick ONE at random. Probability you have the double headed coin: 1/3.
Flip 4 times, all heads. Incidentally, since this is a prerequisite to the question (if it is observed that ...) the probability of this occurring is 100%. You learn nothing about the coin in your hand. Could have two heads, could be fair. Probability your coin is the double headed one: 1/3. It's been a long time since college, but I didn't even have to get my old textbook to figure that one out. |
#8
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Re: A question from prob/stat final
this is incorrect. this first two responses are correct.
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