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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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  • $\begingroup$ It's definitely not true, I'll post a counterexample if no-one else does $\endgroup$ Jul 29, 2014 at 23:18
  • $\begingroup$ Oh unless perhaps you mean for every real number $a$? $\endgroup$ Jul 29, 2014 at 23:19
  • $\begingroup$ Number $a$ is given constant, let's say zero $\endgroup$
    – lowtech
    Jul 29, 2014 at 23:20
  • $\begingroup$ How about "Does there exist a real number a s.t. ..." ? $\endgroup$
    – BCLC
    Jul 29, 2014 at 23:27
  • $\begingroup$ @BCLC I am not sure I understand your hint, could you elaborate pls? $\endgroup$
    – lowtech
    Jul 29, 2014 at 23:43

1 Answer 1

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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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  • $\begingroup$ thanks for this counterexample, i am still struggling with basic intuitions on probabilities... $\endgroup$
    – lowtech
    Jul 29, 2014 at 23:40

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