A question from prob/stat final

[AEKDBet]

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/3215fa05/pdf/final.pdf)

[AaronBrown]

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.

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

[ohnonotthat]

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. /images/graemlins/smirk.gif

[ohnonotthat]

It depends whether this occured live or online since we all know

ONLINE COIN TOSSING IS RIGGED ! /images/graemlins/mad.gif

[AaronBrown]

Thanks for the kind words. I'm tempted to demonstrate the 1%, just to avoid getting a reputation as a wimp.

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.

[AEKDBet]

this is incorrect. this first two responses are correct.