Conditional Probability in a judge/suspect problem
Clash Royale CLAN TAG#URR8PPP
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The verdict given by a judge is 90% reliable when a suspect is guilty and 99% reliable when the suspect is innocent. If the suspect is selected from a group of people of whom 5% have ever committed a crime and the judge finds him guilty, what is the probability that the suspect is innocent?
My job:
Let $I$ the event "the suspect is innocent". Then
$begineqnarray*
P(I)&=&P(I|mboxJudge is reliable)P(mboxJudge is reliable)\
&&+P(I|mboxJudge is not reliable)P(mboxJudge is not reliable)\
endeqnarray*$
On the other hand, I know $P(mboxJudge is reliable|I^c)=0.9$ and $P(mboxJudge is reliable|I)=0.99$. Now, I don't know how to use the bold hypothesis. I don't know how to proced.
probability conditional-probability bayes-theorem
edited Aug 29 at 5:46
David G. Stork
8,10421232
asked Aug 29 at 5:37
sinbadh
6,285724
 |Â
show 4 more comments
up vote
1
down vote
favorite
The verdict given by a judge is 90% reliable when a suspect is guilty and 99% reliable when the suspect is innocent. If the suspect is selected from a group of people of whom 5% have ever committed a crime and the judge finds him guilty, what is the probability that the suspect is innocent?
My job:
Let $I$ the event "the suspect is innocent". Then
$begineqnarray*
P(I)&=&P(I|mboxJudge is reliable)P(mboxJudge is reliable)\
&&+P(I|mboxJudge is not reliable)P(mboxJudge is not reliable)\
endeqnarray*$
On the other hand, I know $P(mboxJudge is reliable|I^c)=0.9$ and $P(mboxJudge is reliable|I)=0.99$. Now, I don't know how to use the bold hypothesis. I don't know how to proced.
probability conditional-probability bayes-theorem
edited Aug 29 at 5:46
David G. Stork
8,10421232
asked Aug 29 at 5:37
sinbadh
6,285724
Use Bayes' theorem
– John Douma
Aug 29 at 5:55
I don't understand why you wrote "ever". Does that imply that the judge's verdict is counted as correct if they find someone guilty who has ever committed a crime, not necessarily the one they were charged with (and innocent only if they've never committed any crime)? Otherwise, the "ever" creates a disconnect between the various data given.
– joriki
Aug 29 at 5:57
1
Define terms of the sort $P[suspect~found~innocent|suspect~IS~innocent]$, $P[suspect~found~guilty|suspect~is~innocent]$, $P[suspect~found~guilty|suspect~IS~guilty]$ and $P[suspect~found~guilty|suspect~is~innocent]$ and work from Bayes rule.
– David G. Stork
Aug 29 at 5:58
Yes. Normally i don't ask for help for this kind of exercises, but i'm really confused in this one.Thanks
– sinbadh
Aug 29 at 6:00
1
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime
– Graham Kemp
Aug 29 at 23:25
 |Â
show 4 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The verdict given by a judge is 90% reliable when a suspect is guilty and 99% reliable when the suspect is innocent. If the suspect is selected from a group of people of whom 5% have ever committed a crime and the judge finds him guilty, what is the probability that the suspect is innocent?
My job:
Let $I$ the event "the suspect is innocent". Then
$begineqnarray*
P(I)&=&P(I|mboxJudge is reliable)P(mboxJudge is reliable)\
&&+P(I|mboxJudge is not reliable)P(mboxJudge is not reliable)\
endeqnarray*$
On the other hand, I know $P(mboxJudge is reliable|I^c)=0.9$ and $P(mboxJudge is reliable|I)=0.99$. Now, I don't know how to use the bold hypothesis. I don't know how to proced.
probability conditional-probability bayes-theorem
edited Aug 29 at 5:46
David G. Stork
8,10421232
asked Aug 29 at 5:37
sinbadh
6,285724
The verdict given by a judge is 90% reliable when a suspect is guilty and 99% reliable when the suspect is innocent. If the suspect is selected from a group of people of whom 5% have ever committed a crime and the judge finds him guilty, what is the probability that the suspect is innocent?
My job:
Let $I$ the event "the suspect is innocent". Then
$begineqnarray*
P(I)&=&P(I|mboxJudge is reliable)P(mboxJudge is reliable)\
&&+P(I|mboxJudge is not reliable)P(mboxJudge is not reliable)\
endeqnarray*$
On the other hand, I know $P(mboxJudge is reliable|I^c)=0.9$ and $P(mboxJudge is reliable|I)=0.99$. Now, I don't know how to use the bold hypothesis. I don't know how to proced.
probability conditional-probability bayes-theorem
edited Aug 29 at 5:46
David G. Stork
8,10421232
asked Aug 29 at 5:37
sinbadh
6,285724
edited Aug 29 at 5:46
David G. Stork
8,10421232
edited Aug 29 at 5:46
David G. Stork
8,10421232
edited Aug 29 at 5:46
David G. Stork
8,10421232
8,10421232
asked Aug 29 at 5:37
sinbadh
6,285724
asked Aug 29 at 5:37
sinbadh
6,285724
asked Aug 29 at 5:37
sinbadh
6,285724
6,285724
Use Bayes' theorem
– John Douma
Aug 29 at 5:55
I don't understand why you wrote "ever". Does that imply that the judge's verdict is counted as correct if they find someone guilty who has ever committed a crime, not necessarily the one they were charged with (and innocent only if they've never committed any crime)? Otherwise, the "ever" creates a disconnect between the various data given.
– joriki
Aug 29 at 5:57
1
Define terms of the sort $P[suspect~found~innocent|suspect~IS~innocent]$, $P[suspect~found~guilty|suspect~is~innocent]$, $P[suspect~found~guilty|suspect~IS~guilty]$ and $P[suspect~found~guilty|suspect~is~innocent]$ and work from Bayes rule.
– David G. Stork
Aug 29 at 5:58
Yes. Normally i don't ask for help for this kind of exercises, but i'm really confused in this one.Thanks
– sinbadh
Aug 29 at 6:00
1
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime
– Graham Kemp
Aug 29 at 23:25
 |Â
show 4 more comments
Use Bayes' theorem
– John Douma
Aug 29 at 5:55
I don't understand why you wrote "ever". Does that imply that the judge's verdict is counted as correct if they find someone guilty who has ever committed a crime, not necessarily the one they were charged with (and innocent only if they've never committed any crime)? Otherwise, the "ever" creates a disconnect between the various data given.
– joriki
Aug 29 at 5:57
1
Define terms of the sort $P[suspect~found~innocent|suspect~IS~innocent]$, $P[suspect~found~guilty|suspect~is~innocent]$, $P[suspect~found~guilty|suspect~IS~guilty]$ and $P[suspect~found~guilty|suspect~is~innocent]$ and work from Bayes rule.
– David G. Stork
Aug 29 at 5:58
Yes. Normally i don't ask for help for this kind of exercises, but i'm really confused in this one.Thanks
– sinbadh
Aug 29 at 6:00
1
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime
– Graham Kemp
Aug 29 at 23:25
Use Bayes' theorem
– John Douma
Aug 29 at 5:55
Use Bayes' theorem
– John Douma
Aug 29 at 5:55
I don't understand why you wrote "ever". Does that imply that the judge's verdict is counted as correct if they find someone guilty who has ever committed a crime, not necessarily the one they were charged with (and innocent only if they've never committed any crime)? Otherwise, the "ever" creates a disconnect between the various data given.
– joriki
Aug 29 at 5:57
I don't understand why you wrote "ever". Does that imply that the judge's verdict is counted as correct if they find someone guilty who has ever committed a crime, not necessarily the one they were charged with (and innocent only if they've never committed any crime)? Otherwise, the "ever" creates a disconnect between the various data given.
– joriki
Aug 29 at 5:57
1
1
Define terms of the sort $P[suspect~found~innocent|suspect~IS~innocent]$, $P[suspect~found~guilty|suspect~is~innocent]$, $P[suspect~found~guilty|suspect~IS~guilty]$ and $P[suspect~found~guilty|suspect~is~innocent]$ and work from Bayes rule.
– David G. Stork
Aug 29 at 5:58
Define terms of the sort $P[suspect~found~innocent|suspect~IS~innocent]$, $P[suspect~found~guilty|suspect~is~innocent]$, $P[suspect~found~guilty|suspect~IS~guilty]$ and $P[suspect~found~guilty|suspect~is~innocent]$ and work from Bayes rule.
– David G. Stork
Aug 29 at 5:58
Yes. Normally i don't ask for help for this kind of exercises, but i'm really confused in this one.Thanks
– sinbadh
Aug 29 at 6:00
Yes. Normally i don't ask for help for this kind of exercises, but i'm really confused in this one.Thanks
– sinbadh
Aug 29 at 6:00
1
1
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime
– Graham Kemp
Aug 29 at 23:25
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime
– Graham Kemp
Aug 29 at 23:25
 |Â
show 4 more comments
2 Answers
2
active
oldest
votes
up vote
0
down vote
This is an application of Bayes' theorem!
$P(I) = P(ImidtextGuilty)P(textGuilty)$
and we know that $P(textGuilty)=0.05$.
Now, using Bayes' theorem,
beginequation
P(ImidtextGuilty) = P(textGuiltymid I) fracP(I)P(textGuilty)
endequation
In our population, the probability of innocent is $0.95$ and the probability of guilty is $0.05$; therefore $P(I)/P(textGuilty)=0.95/0.05$. And $P(textGuiltymid I)=0.01$. So now we can use Bayes' theorem to find $P(ImidtextGuilty)$, and we can plug that in our first equation to get the answer.
edited Aug 29 at 6:06
joriki
167k10180333
answered Aug 29 at 5:59
NicNic8
3,7513922
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
add a comment |Â
up vote
0
down vote
You want the conditional probability that the Suspect is Innocent given the Judge found him Guilty.
The suspect will be found Guilty if either: Is Innocent and Judge not reliable, or Is not Innocent and the Judge is reliable.
$$mathsf P(Imid G)=dfracmathsf P(Icap G)mathsf P(G)=dfracmathsf P(Icap J^complement)mathsf P(Icap J^complement)+mathsf P(I^complementcap J)$$
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
This is an application of Bayes' theorem!
$P(I) = P(ImidtextGuilty)P(textGuilty)$
and we know that $P(textGuilty)=0.05$.
Now, using Bayes' theorem,
beginequation
P(ImidtextGuilty) = P(textGuiltymid I) fracP(I)P(textGuilty)
endequation
In our population, the probability of innocent is $0.95$ and the probability of guilty is $0.05$; therefore $P(I)/P(textGuilty)=0.95/0.05$. And $P(textGuiltymid I)=0.01$. So now we can use Bayes' theorem to find $P(ImidtextGuilty)$, and we can plug that in our first equation to get the answer.
edited Aug 29 at 6:06
joriki
167k10180333
answered Aug 29 at 5:59
NicNic8
3,7513922
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
add a comment |Â
up vote
0
down vote
This is an application of Bayes' theorem!
$P(I) = P(ImidtextGuilty)P(textGuilty)$
and we know that $P(textGuilty)=0.05$.
Now, using Bayes' theorem,
beginequation
P(ImidtextGuilty) = P(textGuiltymid I) fracP(I)P(textGuilty)
endequation
In our population, the probability of innocent is $0.95$ and the probability of guilty is $0.05$; therefore $P(I)/P(textGuilty)=0.95/0.05$. And $P(textGuiltymid I)=0.01$. So now we can use Bayes' theorem to find $P(ImidtextGuilty)$, and we can plug that in our first equation to get the answer.
edited Aug 29 at 6:06
joriki
167k10180333
answered Aug 29 at 5:59
NicNic8
3,7513922
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
add a comment |Â
up vote
0
down vote
up vote
0
down vote
This is an application of Bayes' theorem!
$P(I) = P(ImidtextGuilty)P(textGuilty)$
and we know that $P(textGuilty)=0.05$.
Now, using Bayes' theorem,
beginequation
P(ImidtextGuilty) = P(textGuiltymid I) fracP(I)P(textGuilty)
endequation
In our population, the probability of innocent is $0.95$ and the probability of guilty is $0.05$; therefore $P(I)/P(textGuilty)=0.95/0.05$. And $P(textGuiltymid I)=0.01$. So now we can use Bayes' theorem to find $P(ImidtextGuilty)$, and we can plug that in our first equation to get the answer.
edited Aug 29 at 6:06
joriki
167k10180333
answered Aug 29 at 5:59
NicNic8
3,7513922
This is an application of Bayes' theorem!
$P(I) = P(ImidtextGuilty)P(textGuilty)$
and we know that $P(textGuilty)=0.05$.
Now, using Bayes' theorem,
beginequation
P(ImidtextGuilty) = P(textGuiltymid I) fracP(I)P(textGuilty)
endequation
In our population, the probability of innocent is $0.95$ and the probability of guilty is $0.05$; therefore $P(I)/P(textGuilty)=0.95/0.05$. And $P(textGuiltymid I)=0.01$. So now we can use Bayes' theorem to find $P(ImidtextGuilty)$, and we can plug that in our first equation to get the answer.
edited Aug 29 at 6:06
joriki
167k10180333
answered Aug 29 at 5:59
NicNic8
3,7513922
edited Aug 29 at 6:06
joriki
167k10180333
edited Aug 29 at 6:06
joriki
167k10180333
edited Aug 29 at 6:06
joriki
167k10180333
167k10180333
answered Aug 29 at 5:59
NicNic8
3,7513922
answered Aug 29 at 5:59
NicNic8
3,7513922
answered Aug 29 at 5:59
NicNic8
3,7513922
3,7513922
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
add a comment |Â
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
There is a distinction between being guilty and being found guilty. We are told $5%$ of the population have committed a crime, not that the judge will find $5%$ of them guilty.
– Graham Kemp
Aug 29 at 23:30
add a comment |Â
up vote
0
down vote
You want the conditional probability that the Suspect is Innocent given the Judge found him Guilty.
The suspect will be found Guilty if either: Is Innocent and Judge not reliable, or Is not Innocent and the Judge is reliable.
$$mathsf P(Imid G)=dfracmathsf P(Icap G)mathsf P(G)=dfracmathsf P(Icap J^complement)mathsf P(Icap J^complement)+mathsf P(I^complementcap J)$$
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
add a comment |Â
up vote
0
down vote
You want the conditional probability that the Suspect is Innocent given the Judge found him Guilty.
The suspect will be found Guilty if either: Is Innocent and Judge not reliable, or Is not Innocent and the Judge is reliable.
$$mathsf P(Imid G)=dfracmathsf P(Icap G)mathsf P(G)=dfracmathsf P(Icap J^complement)mathsf P(Icap J^complement)+mathsf P(I^complementcap J)$$
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You want the conditional probability that the Suspect is Innocent given the Judge found him Guilty.
The suspect will be found Guilty if either: Is Innocent and Judge not reliable, or Is not Innocent and the Judge is reliable.
$$mathsf P(Imid G)=dfracmathsf P(Icap G)mathsf P(G)=dfracmathsf P(Icap J^complement)mathsf P(Icap J^complement)+mathsf P(I^complementcap J)$$
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
You want the conditional probability that the Suspect is Innocent given the Judge found him Guilty.
The suspect will be found Guilty if either: Is Innocent and Judge not reliable, or Is not Innocent and the Judge is reliable.
$$mathsf P(Imid G)=dfracmathsf P(Icap G)mathsf P(G)=dfracmathsf P(Icap J^complement)mathsf P(Icap J^complement)+mathsf P(I^complementcap J)$$
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
answered Aug 29 at 6:08
Graham Kemp
81.1k43275
81.1k43275
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
add a comment |Â
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime.
– Graham Kemp
Aug 29 at 23:26
add a comment |Â
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Use Bayes' theorem
– John Douma
Aug 29 at 5:55
I don't understand why you wrote "ever". Does that imply that the judge's verdict is counted as correct if they find someone guilty who has ever committed a crime, not necessarily the one they were charged with (and innocent only if they've never committed any crime)? Otherwise, the "ever" creates a disconnect between the various data given.
– joriki
Aug 29 at 5:57
1
Define terms of the sort $P[suspect~found~innocent|suspect~IS~innocent]$, $P[suspect~found~guilty|suspect~is~innocent]$, $P[suspect~found~guilty|suspect~IS~guilty]$ and $P[suspect~found~guilty|suspect~is~innocent]$ and work from Bayes rule.
– David G. Stork
Aug 29 at 5:58
Yes. Normally i don't ask for help for this kind of exercises, but i'm really confused in this one.Thanks
– sinbadh
Aug 29 at 6:00
1
Of course, one issue here is that we are only told that the suspect come from a group where $5%$ have ever committed a crime; not necessarily this crime
– Graham Kemp
Aug 29 at 23:25