A plane is missing, and it is presumed that it was equally likely to have gone down in any of 3 possible regions. If the plane is in a given region, the conditional probability that the plane will be found upon a search is 0.6 for region 1, 0.7 for region 2 and 0.8 for region 3. Given that a search in region 1 is unsuccessful, what is the conditional probability (up to 2 decimals) that the plane is in region 1?

I think that I should use the Bayer's Law, the result should be 0.17. Can someone explain to me how can I compute the two probabilities that I need, if my method is correct? Thanks in advance! P(1)=P(2)=P(3)=1/3
P($F_1^c$|1) = P($F^c$|1)
P($F^c$∣1)=0.4 (1-0.6)
P($F^c$∣2)=0.3 (1-0.7)
P($F^c$∣3)=0.2 (1-0.8)

I don't know how to compute $P(F_1^c∣2)$ and $P(F_1^c∣3)$

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    $\begingroup$ Welcome to math.SE. Please see this tutorial and reference on how to typeset math on this site. $\endgroup$ – joriki Sep 18 '18 at 8:23
  • $\begingroup$ Can someone please help me? Thanks $\endgroup$ – TFAE Sep 20 '18 at 17:13
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    $\begingroup$ I updated my answer in response to your edit. Did you check out the tutorial/reference I linked to above? $\endgroup$ – joriki Sep 20 '18 at 17:29
  • $\begingroup$ Yes, I checked it, thanks, now I understand how to write in math mode! $\endgroup$ – TFAE Sep 20 '18 at 17:37

A search in region $1$ is unsuccessful. No information about searches in the other regions is provided. Your notation doesn't distinguish between a search failure in region $1$ and an overall search failure. That's OK in the case $\mathsf P(\overline F\mid 1)=\mathsf P(\overline F_1\mid 1)$, since conditional on the plane being in region $1$, the search fails exactly if it fails in region $1$; but the same isn't true in the case $\mathsf P(1\mid\overline F)\neq\mathsf P(1\mid\overline F_1)$.

Update in response to the edit in the question:

$P(\overline{F_1}\mid2)$ is the probability that a search in region $1$ fails, given that the plane is in region $2$. That probability is $1$.

  • $\begingroup$ So what I should do? How does my overall notation change? $\endgroup$ – TFAE Sep 18 '18 at 15:49
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    $\begingroup$ Thank you very much for the answer, I understood where I was wrong and how to fix it! Thanks, have a nice day! $\endgroup$ – TFAE Sep 20 '18 at 17:38

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