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
5 years, 11 months ago

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4 votes

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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 2/32/3 (chance that A wrote '+') times the sum of (2/3)3(2/3)^3 and 3(1/3)22/33\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 28/8128/81.

The second has chance 1/31/3 (chance that A wrote '-') times the sum of (1/3)3(1/3)^3 and 3(2/3)21/33\cdot(2/3)^2 \cdot 1/3 (chance that it was changed 3 times or 1 time). This chance is 12/8112 / 81.

Thus, the total chance that the referee saw '+' is 40/8140/81. The chance that the referee saw '+' and A wrote '+' is 28/8128/81, so the chance that A wrote '+' given the referee saw '+' is (28/81)/(40/81)=7/10(28 / 81) / (40 / 81) = 7 / 10.(Note the general identity P(A and B)=P(A)P(BA)P(A \ \mathrm{and}\ B) = P(A) \cdot P(B | A), in this case AA is that the referee saw '+' and BB 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 1/21/2, the sum cannot be collapsed due to symmetry. If there are nn middlemen (in this case, n=3n = 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(2/3 + 1/3)^n. The eventual answer will be the quotient of 2/32/3 times the even change chance and the sum of 2/32/3 times the even change chance plus 1/31/3 times the odd change chance.

Alexander Bourzutschky - 5 years, 11 months ago

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Thanks Alexander, thanks a lot!........My solution was HIGHLY incorrect!!

Kulkul Chatterjee - 5 years, 11 months ago

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Good question man............

Jit Ganguly - 5 years, 11 months ago

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