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I am probably misusing Bayes' theorem, but I can't figure out how.

$$P(B|A) = 0.4$$

$$P(B) = 0.1$$

$$P(A) = 0.3$$

$$P(A|B) = \dfrac{0.4 \cdot 0.3}{0.1} = 1.2$$

Since the probability of anything can't be > $1$, I must be wrong somewhere. Please help.

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up vote 8 down vote accepted

It is not possible to have the hypothesized probabilities. $P(B|A)=\frac{P(A\cap B)}{P(A)}\leq\frac{P(B)}{P(A)}=\frac{1}{3}\lt 0.4$.

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Thank you! I see where I went wrong mathematically, but do you have any suggestions on developing the intuition for this? For the problem that I was solving, I picked the probabilities intuitively. Is there a way to quickly check that the probabilities I picked make sense? (Because they make intuitive sense.) –  Alexei Andreev Jun 18 '11 at 19:53
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@Alexei: To me, $P(B|A)=\frac{P(A\cap B)}{P(A)}$ and $P(A\cap B)\leq P(B)$ are intuitive, and these are all that is needed to check the plausibility here. You can quickly check that $P(B)\geq P(A)P(B|A)$. But I don't know about developing intuition as it applies to the problem you were trying to solve, because I have no idea what that problem is. –  Jonas Meyer Jun 18 '11 at 20:13
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