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For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.
Standard notation:
probability of A given B is written \( \Pr(A \mid B) \)
probability of B is written \( \Pr(B) \)
Bayes' Theorem:
Using the notation above, Bayes' Theorem can be written: \[ \Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)} \]Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.
Assume we pick Door 1 and then Monty shows us a goat behind Door 2. Now let A be the event that the car is behind Door 1 and B be the event that Monty shows us a goat behind Door 2. Then
\begin{aligned}
\Pr (A \mid B) &= \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)} \\
&= \frac{1/2\times 1/3}{1/3\times 1/2+1/3\times 0+1/3\times 1} \\
&= 1/3.
\end{aligned}The tricky calculation is \( \Pr(B) \). Remember, we are assuming we initially chose Door 1. It follows that if the car is behind Door 1, Monty will show us a goat behind Door 2 half the time. If the car is behind Door 2, Monty never shows us a goat behind Door 2. Finally, if the car is behind Door 3, Monty shows us a goat behind Door 2 every time. Thus, \[ \Pr(B) = 1/3\times 1/2+1/3\times 0+1/3\times 1 = 1/2. \]The car is either behind Door 1 or Door 3, and since the probability that it's behind Door 1 is 1/3 and the sum of the two probabilities must equal 1, the probability the car is behind Door 3 is \( 1-1/3 = 2/3 \). You could also apply Bayes' Theorem directly, but this is simpler.
So Bayes says we should switch, as our probability of winning the car jumps from 1/3 to 2/3.
Standard notation:
probability of A given B is written \( \Pr(A \mid B) \)
probability of B is written \( \Pr(B) \)
Bayes' Theorem:
Using the notation above, Bayes' Theorem can be written: \[ \Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)} \]Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.
Assume we pick Door 1 and then Monty shows us a goat behind Door 2. Now let A be the event that the car is behind Door 1 and B be the event that Monty shows us a goat behind Door 2. Then
\begin{aligned}
\Pr (A \mid B) &= \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)} \\
&= \frac{1/2\times 1/3}{1/3\times 1/2+1/3\times 0+1/3\times 1} \\
&= 1/3.
\end{aligned}The tricky calculation is \( \Pr(B) \). Remember, we are assuming we initially chose Door 1. It follows that if the car is behind Door 1, Monty will show us a goat behind Door 2 half the time. If the car is behind Door 2, Monty never shows us a goat behind Door 2. Finally, if the car is behind Door 3, Monty shows us a goat behind Door 2 every time. Thus, \[ \Pr(B) = 1/3\times 1/2+1/3\times 0+1/3\times 1 = 1/2. \]The car is either behind Door 1 or Door 3, and since the probability that it's behind Door 1 is 1/3 and the sum of the two probabilities must equal 1, the probability the car is behind Door 3 is \( 1-1/3 = 2/3 \). You could also apply Bayes' Theorem directly, but this is simpler.
So Bayes says we should switch, as our probability of winning the car jumps from 1/3 to 2/3.
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ReplyDeleteDoesn't Pr(B) = 1/2?
ReplyDelete[Pr(B) = 1/3 x 1/2 + 1/3 x 0 + 1/3 x 1 = 1/2]
Hi John! You're correct, of course. I'll correct these calculations now.
ReplyDeleteHow to you get Pr(BIA) = 1/2?
ReplyDeleteThx
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Delete@Mare
ReplyDeletePr(B|A) is the probability Monty opens Door2 given the car is behind Door1 (the door you picked). Since Monty has a choice of 2 goat doors to open in this scenario, the probability he opens Door2 is 1/2.
Given your interest in the quarks, may I ask if you have a view on Bayesian interpretations of quantum theory, such as QBism (http://en.wikipedia.org/wiki/QBism)? As an interested non-mathematician, I'd love to read an explanation of how QBIsm resolves "spooky" quantum theory features such as action-at-a-distance and many-state-reality, especially in the context of developments like quantum key distribution or quantum computing, which appear to use those exactly those features.
ReplyDeleteStatistical/Math Bayes has nothing in common with QBism except for the appellation.
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ReplyDeleteYour error is here: "and since the probability that it's behind Door 1 is 1/3 ".
ReplyDeleteThis is not true. The probability that it's behind Door 1 changes to 1/2 when a third of the probabilities are eliminated by opening Door 2.
Talk about mass hypnosis.
"The probability that it's behind Door 1 changes to 1/2 ..."
DeleteI would be interested to see a Bayesian explanation for that.
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ReplyDeleteThis comment has been removed by the author.
ReplyDeleteHi Christopher,
ReplyDeleteIf we define C as Contestant selects Door #1
Isn't the value you calculated as Pr(B) in actuality Pr(B|C)? Shouldn't we use Pr(B) without condition C?
Regards,
Tom
This is a rather strange and unnecessary use of Bayes' theorem.
ReplyDeleteYou really didn't need Bayes to conclude that the probability of the car being behind the door you (initially) picked is 1/3. This is given! And in fact you use this same probability as "P(A)".
The fact that Monty does "B" doesn't matter to event "A" (there's either a car behind door 1 or not, no matter what Monty does afterwards (!) with the remaining doors), so "P(A|B)" and "P(A)" must always be equal here.
So your explanation of the Monty Hall problem really just starts being interesting/correct and addressing the problem at
"The car is either behind Door 1 or Door 3, and since ..."
quite an easy puzzle
ReplyDeletequite an easy puzzle
ReplyDelete