Comp Sci What is the probability that there is a burglary given John and Mary calls?

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The discussion revolves around calculating the probability of a burglary given calls from John and Mary using Bayes' Theorem. Participants express confusion over the tutorial video and the steps involved in solving the problem. Key points include the need to determine the independence of events, specifically whether John and Mary’s calls are independent and how they relate to the alarm event. Different scenarios are proposed to illustrate how dependencies can affect the calculations, leading to varying results for the probability. Ultimately, it is suggested that if independence is assumed, the probability can be calculated as 0.665, which lies between the extremes derived from the two cases discussed.
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Homework Statement
What is the probability that there is a burglary given John and Mary calls?
Relevant Equations
bayes theorem maybe.
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this is the question



Here is a tutorial video but his steps are very confusing to me. I personally know bayes theorem and have already studied probability and got good marks in it(It may not be a metric for being quality in it given that it is nepal we are talking about.)
https://courses.engr.illinois.edu/ece448/sp2020/slides/lec15.pdf
here is the slide I'm referring to. The answer seems 0.72 or 0.28 according to video.

My attempt-:
I try finding P(B/(J,M)) but I don't get a way to find it. Burglary is independent of anything else. IDK how to find this value.
 
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Try breaking it into parts, eg:

P(B | J & M) = P(B | A) x P(A | J & M)
Then use Bayes' Theorem to work out P(B | A) and P(A | J & M).

Note that the question gives you P(A | B & E), P(A | B & ~E), P(~B & E), P(~B & ~E)
[left-side table, ~ means NOT]
plus P(B), P(E) and since you said they're independent, you have P(B & E) = P(B)P(E)
[top two values]
plus P(J | A), P(J | ~A), P(M | A), P(M | ~A)
[bottom of slide]

You are going to need the value for P(J & M) which means you need to know whether J and M are independent ( in which case we'll have P(J & M) = P(J) P(M) ). Do they tell you that?

You are also going to need the value for P(J & M | A) which means you need to know whether J|A and M|A are independent ( in which case we'll have P(J & M | A) = P(J | A) P(M | A) ). Do they tell you that?
 
andrewkirk said:
Try breaking it into parts, eg:

P(B | J & M) = P(B | A) x P(A | J & M)
Then use Bayes' Theorem to work out P(B | A) and P(A | J & M).

Note that the question gives you P(A | B & E), P(A | B & ~E), P(~B & E), P(~B & ~E)
[left-side table, ~ means NOT]
plus P(B), P(E) and since you said they're independent, you have P(B & E) = P(B)P(E)
[top two values]
plus P(J | A), P(J | ~A), P(M | A), P(M | ~A)
[bottom of slide]
..
andrewkirk said:
You are going to need the value for P(J & M) which means you need to know whether J and M are independent ( in which case we'll have P(J & M) = P(J) P(M) ). Do they tell you that?
Yes J and M are independent to each other.
andrewkirk said:
You are also going to need the value for P(J & M | A) which means you need to know whether J|A and M|A are independent ( in which case we'll have P(J & M | A) = P(J | A) P(M | A) ). Do they tell you that?
J is dependent on A and so is M.
 
shivajikobardan said:
J is dependent on A and so is M.
Yes we know that, but that's not enough, as it still leaves different possibilities. We need to know P(J | M&A) and P(M | J & A).
Consider the following two cases:

Case 1:
A & M =>J (If the alarm goes off and Mary calls then John always calls too)
P(J | M & A) = 1
So P(J & M | A) = P(J&M&A) / P(A) = P(J | M&A) P(M&A) /P(A) = P(M&A) / P(A) = P(M|A)P(A)/P(A) = P(M|A) = 0.7

Case 2:
A & ~M => J (If alarm goes off and Mary doesn't call then John does call)
P(J | A & ~M) = 1

So P(J & ~M & A) = P(J | A & ~M) P(A & ~M) = P(J | A & ~M) P(~M | A) P(A) = 1 x (1 - 0.7) P(A) = 0.3 P(A)
So P(J & M & A) = P(J & A) - P(J & ~M & A) = P(J | A) P(A) - 0.3P(A) = 0.95 P(A) - 0.3 P(A) = 0.65 P(A)
So P(J & M | A) = 0.65 P(A) / P(A) = 0.65.

So these different dependencies give us different results.

If they don't give you any information about those dependencies, I expect they intended - but forgot to say - that the events M | A and J | A are independent, so that P(J & M | A) = P(J | A) P(M | A) = 0.95 x 0.7 = 0.665. Note how that is between the values from the above two cases.
 

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