0
$\begingroup$

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)=0.6
P(F∣2)=0.7
P(F∣3)=0.8
P(1|$F_1^c$)=?
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)
P(1∣$F_1^c$)=$\frac{P(F_1^c∣1)P(1)}{P(F_1^c∣1)P(1)+P(F_1^c∣2)P(2)+P(F_1^c∣3)P(3)}$=$\frac{0.4/3}{0.4/3+P(F_1^c∣2)/3+P(F_1^c∣3)/3}$=??

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

$\endgroup$
4
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ 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
1
$\begingroup$

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 $1$.

$\endgroup$
2
  • $\begingroup$ So what I should do? How does my overall notation change? $\endgroup$ – TFAE Sep 18 '18 at 15:49
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.