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E and F are independent iff $\frac{p(E \cap F)}{p(F)}=p(E)$.

Also, E and F are independent iff $p(E | F)=p(E)$

Why do I get nonsense if I combine the two? I'd get: E and F are independent iff $p(E | F) = \frac{p(E \cap F)}{p(F)}$

Why doesn't the iff carry over when I combine the two equations?

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    $\begingroup$ Why would you expect that to work? $\endgroup$ – Eric Wofsey Mar 4 '18 at 6:28
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    $\begingroup$ @EricWofsey I consider iff to indicate equality, so I was intuitively expecting the "iff" to carry over and I'm not grasping why it isn't $\endgroup$ – Goldname Mar 4 '18 at 6:29
  • $\begingroup$ You have combined illogically. What we do have is : $E, F $ are independent iff $p(E)=p(E|F)=p(E\cap F)/p(F).$.... For $any $ $ E $ we have $ p(E|F)=p(E\cap F)/p(F)$ so $E,F$ are independent iff $p(E) $ is equal to either of them. $\endgroup$ – DanielWainfleet Mar 5 '18 at 9:42
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Let $a=\frac{p(E \cap F)}{p(F)}$, $b=p(E | F)$, and $c=p(E)$. You know that the equations $a=c$ and $b=c$ are equivalent to each other (since they are both equivalent to "$E$ and $F$ are independent"). That is, whenever $a=c$ is true, $b=c$ is also true, and conversely.

There is no reason for $a=b$ to be equivalent to these two statements, though. If $a=c$ is true, then $b=c$ is also true, and so $a=b$ is true as well. So we have an implication in one direction. But the other direction doesn't work: if we know $a=b$, that doesn't tell us anything about whether $a=c$ or $b=c$ are true. It's entirely possible that $a$ and $b$ are equal to each other, but not to $c$.

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    $\begingroup$ In fact, $a$ and $b$ in this case are equal to each other by definition, regardless of whether $E$ and $F$ are independent. $\endgroup$ – Bolton Bailey Mar 5 '18 at 3:48
  • $\begingroup$ What you did is equivalent to saying, "E and F are independent if E and F are independent." $\endgroup$ – richard1941 Mar 12 '18 at 15:37
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You combined them wrong. From $A \Leftrightarrow B$ and $A \Leftrightarrow C$, you can infer $B \Leftrightarrow C$.

But you've misidentified what $A,B,C$ are. They aren't the sides of the equation, but rather

  • $A$ is the statement "$E$ and $F$ are independent"
  • $B$ is the formula $\frac{p(E \cap F)}{p(F)}=p(E)$
  • $C$ is the formula $p(E | F)=p(E)$

so the correct combination is

$$ \frac{p(E \cap F)}{p(F)}=p(E) \qquad \mathrm{iff} \qquad p(E | F)=p(E)$$

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Here is a simple mathematical example to clearly elucidate the logical error:

An integer $n$ is an even prime iff $n = 2$.

An integer $n$ is an even prime iff $n = 1+1$.

(WRONG!) An integer $n$ is an even prime iff $2 = 1+1$.

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    $\begingroup$ I don't feel like this example is much different than my question, so it doesn't elucidate anything for me. $\endgroup$ – Goldname Mar 5 '18 at 19:44
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[This answer assumes the OP's argument is different from that assumed by the other answers.]

There's a logical fallacy in your argument, which seems to be of the following form:

  1. $A\iff(x=y)$
  2. $A\iff(z=y)$, where we define $z::=x$
  3. Hence, $A\iff [(x=y)\land (z=y)]$

So far, so good, but the next step, eliminating $y$ in the conjunction, is incorrect:

  1. Therefore, $A\iff(x=z)$.

That step would be correct if $$[(x=y)\land (z=y)]\ \ \iff\ \ (x=z), $$ but this fails simply because $$[(x=y)\land (z=y)]\require{cancel}\ \ \cancel\impliedby\ \ (x=z). $$

NB: In your case, $A$=("$E,F$ are independent"), $x=\frac{p(E \cap F)}{p(F)},$ $y=p(E),$ and $z=p(E\mid F).$

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To address the more general question, here’s an example:

Suppose Kim Jong Un tells you, “In a democracy, what the leader promised is what the people voted for, and what the people voted for is what the government does. Just bear with me and take that as a definition. Therefore, I rule a democracy, because what the leader promised is what the government does.”

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