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Probability Question

Hi people! I'm new to probability and so while going through conditional probability I found the following problem which I solved by arguing directly without using any results of Conditional Probability ............unfortunately the answer wasn't given. The problem goes as:

There are 4 friends A,B,C, D and a referee. A puts a '+' or a '-' sign on a piece of paper and passes it to B who perhaps changes the sign and passes the slip to C, who may change the sign and passes it to D. D perhaps changes the sign and finally passes the slip to the referee. Suppose 2/3 is the prob. of writing a '+' by A and each of B,C,D changes the sign by prob 1/3 independently. If the referee notices a '+' sign on the slip, then find the prob that A wrote a '+' on the slip.

I would really appreciate if someone gave me the solution. Also assuming the same conditions, what would then be the ans, instead of 4 friends if n friends did the same thing.

Note by Kulkul Chatterjee
3 years, 10 months ago

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This is my first impression of how to solve it. I'm not sure my numbers are all right, but I'm fairly sure this should be the approach:

The chance that a referee saw '+' is the chance that it was '+' originally and was changed 0 or 2 times, or the chance that it was '-' originally and was changed 1 or 3 times.

The first of these has chance \(2/3\) (chance that A wrote '+') times the sum of \((2/3)^3\) and \(3\cdot(1/3)^2\cdot 2/3\) (chance that it was changed 0 times or 2 times). Thus, the chance that A wrote '+' and the referee saw '+' is equal to \(28/81\).

The second has chance \(1/3\) (chance that A wrote '-') times the sum of \((1/3)^3\) and \(3\cdot(2/3)^2 \cdot 1/3\) (chance that it was changed 3 times or 1 time). This chance is \(12 / 81\).

Thus, the total chance that the referee saw '+' is \(40/81\). The chance that the referee saw '+' and A wrote '+' is \(28/81\), so the chance that A wrote '+' given the referee saw '+' is \((28 / 81) / (40 / 81) = 7 / 10\).(Note the general identity \(P(A \ \mathrm{and}\ B) = P(A) \cdot P(B | A)\), in this case \(A\) is that the referee saw '+' and \(B\) is the chance that A wrote '+.')

To generalize this, note that the chance that the number of changes is odd is 1 minus the chance that the number of changes is even. The only difficulty is calculating one without knowing the other. Since the chance of changing is not \(1/2\), the sum cannot be collapsed due to symmetry. If there are \(n\) middlemen (in this case, \(n = 3\)), then the chance of no changes is given by the first, third, etc. terms in the binomial expansion of \((2/3 + 1/3)^n\). The eventual answer will be the quotient of \(2/3\) times the even change chance and the sum of \(2/3\) times the even change chance plus \(1/3\) times the odd change chance. Alexander Bourzutschky · 3 years, 9 months ago

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@Alexander Bourzutschky Thanks Alexander, thanks a lot!........My solution was HIGHLY incorrect!! Kulkul Chatterjee · 3 years, 9 months ago

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Good question man............ Jit Ganguly · 3 years, 9 months ago

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