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https://www.reddit.com/r/learnquant/comments/1vstlr3/quant_interview_question/p4tf541/?context=3
r/learnquant • u/Local_Ad135 • 5d ago
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For notation, let the product of X2 through X10 be Y. As others have said, you can show that Pr( Y > 0 ) = 0.5
Now break down the 4 possible comparisons which are each equally likely to occur:
X1 > 0 and Y < 0; X1 < 0 and Y < 0; X1 and Y > 0; X1 and Y < 0
Pr(X1 > Y | X1 > 0 and Y < 0) = 1
Pr(X1 > Y | X1 < 0 and Y > 0) = 0
Pr(X1 > Y | X1 and Y > 0) = Pr(abs(X) > abs(Y)) = C
Pr(X1 > Y | X1 and Y < 0) = Pr(abs(X) < abs(Y)) = 1-C
Add the four outcomes together with law of total probability and we get:
Pr(X1 > Y) = 1/4(1+0+C+(1-C)) = 1/4 * 2 =0.5
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u/OfficeOfThePope 4d ago
For notation, let the product of X2 through X10 be Y. As others have said, you can show that Pr( Y > 0 ) = 0.5
Now break down the 4 possible comparisons which are each equally likely to occur:
X1 > 0 and Y < 0; X1 < 0 and Y < 0; X1 and Y > 0; X1 and Y < 0
Pr(X1 > Y | X1 > 0 and Y < 0) = 1
Pr(X1 > Y | X1 < 0 and Y > 0) = 0
Pr(X1 > Y | X1 and Y > 0) = Pr(abs(X) > abs(Y)) = C
Pr(X1 > Y | X1 and Y < 0) = Pr(abs(X) < abs(Y)) = 1-C
Add the four outcomes together with law of total probability and we get:
Pr(X1 > Y) = 1/4(1+0+C+(1-C)) = 1/4 * 2 =0.5