Given some real number $a$ can anyone prove that if $$ P(X > a) > P(Y > a) $$ is true then $$ P(X > Y) > \frac12 $$ is also true.
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It's definitely not true, I'll post a counterexample if no-one else does – John Fernley Jul 29 '14 at 23:18
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Oh unless perhaps you mean for every real number $a$? – John Fernley Jul 29 '14 at 23:19
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Number $a$ is given constant, let's say zero – lowtech Jul 29 '14 at 23:20
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How about "Does there exist a real number a s.t. ..." ? – BCLC Jul 29 '14 at 23:27
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@BCLC I am not sure I understand your hint, could you elaborate pls? – lowtech Jul 29 '14 at 23:43
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@lowtech Oh sorry it's not a hint. I was a suggesting another question. What if the 0.6 in T.Bongers' counterexample was lower? That is, how low does 0.6 have to brought down for the given X and Y in order for P(X>Y) > 0.5 to hold? – BCLC Jul 29 '14 at 23:44
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So does there exist a real number a s.t. P(X>a) > P(Y>a) implies P(X>Y)>0.5? – BCLC Jul 29 '14 at 23:45
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$X$ takes value $1$ with probability $0.01$ and $0$ with probability $.99$.
$Y$ takes value $.5$ with probability $1$.
Then $P(X > .6) > 0 = P(Y > .6)$, while $P(X > Y)$ is $.01 < 1/2$.
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thanks for this counterexample, i am still struggling with basic intuitions on probabilities... – lowtech Jul 29 '14 at 23:40