Conditional probability (does anyone know what to condition this on?)

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joebadcock1

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I am trying to answer this question but am getting confused, if anyone can help me that would be great.

​
First appeal ​
Second appeal ​
Outcome​
Successful​
No second appeal​
Entrance allowed​
Rejected​
Successful​
Entrance allowed​
Rejected​
Rejected​
Rejected​
​

Historic data shows that 70% of people who end up in the detention hotel, are there because they made a minor mistake in the visa application, while 10% get put into the detention hotel because they lied in the visa application (and it’s possible to prove it), while 20% have missing documents. If a minor mistake was the reason behind the initial rejection, then each appeal has a 95% chance of being successful. However if the reason was a provable lie or missing documents, then an appeal gets rejected with probability 0.93. (a) [7 marks] Considering only the first round of appeals, find the probability that a randomly chosen person in the detention hotel made a minor mistake in their application given that their first appeal is approved.

Q = Considering only the first round of appeals, find the probability that a randomly chosen person in the detention hotel made a minor mistake in their application given that their first appeal is approved?

So far i have 0.7 x 0.95% = 0.665 which is the probability that a person with missing documents get in. However i'm not sure what this then needs to be conditioned on given the first appeal? or is it a trick question and this is the answer
 
The goal is to find
[math]\Pr(\text{minor mistake}|\text{success}) =\frac{\Pr(\text{success}|\text{minor mistake})\Pr(\text{minor mistake})}{\Pr(\text{success})} =\frac{0.95(0.7)}{\Pr(\text{success})}[/math] Next, find [imath]\Pr(\text{success})[/imath], the marginal probability of a successful appeal i.e. non-conditional.
 
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joebadcock1 said:
Yeh thats what im stuck with, how do i find the probability of a successful appeal?
Click to expand...
Please share your thoughts regarding the course of action? What does your class-notes say? What does your textbook say?
 
I am thinking, that would be the probability of being succesful regardless of the reason the person is in the room in first place.
Pr(sucess) = 0.95 + 0.07
 
joebadcock1 said:
I am thinking, that would be the probability of being succesful regardless of the reason the person is in the room in first place.
Pr(sucess) = 0.95 + 0.07
Click to expand...
Does Pr mean probability?

0.95 + 0.07 = 1.02 > 1.......... Probabilities CANNOT be more than 1.
 
joebadcock1 said:
I am thinking, that would be the probability of being succesful regardless of the reason the person is in the room in first place.
Pr(sucess) = 0.95 + 0.07
Click to expand...
That violates a fundamental axiom of probability. The probability of any event cannot be greater than 1. Does the law of total probability ring a bell?
 
Yes, i did think that but am lost in how to work it out, and we cant multiply it as they happen at the same time. I have heard of the law of total probability but would be unsure of how to use it to find P(success) in this case
 
So using the laws of total probability the probabliity of a success is 0.686. So to answer the question, 0.95 x 0.7 x 0.686 = 0.9694
 
joebadcock1 said:
So using the laws of total probability the probabliity of a success is 0.686. So to answer the question, 0.95 x 0.7 x 0.686 = 0.9694
Click to expand...
Try again. If you are multiplying positive numbers, each of which are less than 1 - the product must be less than the minimum of the factors.
 
Sorry made a typo, 0.95 x 0.7 / 0.686 = 0.966

So to find the probability of a success I used total probability to find 0.665 + 0.021 = 0.686. Then i plugged this number into the formula as above. Is this not correct, if not which part is wrong?
 
Read this thread, specifically post #6. Apply the same theories to your question:

The Cigarette Question

The sample includes 30% of women and 70% of men. It is known that among them 10% of women and 60% of men smoke. Person X is known to be a non-smoker. Find the probability that X is male.
www.freemathhelp.com
 
Yeh thats what I already have.

We know;

P(MM) = 0.70, P(Success|MM) = 0.95

The goal is to find the minor mistake given that the appeal is successful

P(MM | Appeal is successful) = P(MM ∩ appeal is successful) /
P(Appeal is successful)

The using the law of total probability we can do 0.7 x 0.95 = 0.665 for a missing mistake success and 0.3 + 0.07 for a L/MDoc success = 0.686

P(MM | Appeal is successful) = 0.7 x 0.95 = 0.665 / 0.686 = 0.966

So the answer of a
P(MM | Appeal is successful) = 0.966

Not too sure how I'm going wrong here?
 
joebadcock1 said:
Yeh thats what I already have.

We know;

P(MM) = 0.70, P(Success|MM) = 0.95

The goal is to find the minor mistake given that the appeal is successful

P(MM | Appeal is successful) = P(MM ∩ appeal is successful) /
P(Appeal is successful)

The using the law of total probability we can do 0.7 x 0.95 = 0.665 for a missing mistake success and 0.3 + 0.07 for a L/MDoc success = 0.686

P(MM | Appeal is successful) = 0.7 x 0.95 = 0.665 / 0.686 = 0.966

So the answer of a
P(MM | Appeal is successful) = 0.966

Not too sure how I'm going wrong here?
Click to expand...
Using Law of Total Probability:
\(\displaystyle \Pr(\text{success})=\Pr(\text{sucess} \cap \text{mistake})+\Pr(\text{sucess} \cap\text{lied}) + \Pr(\text{sucess} \cap \text{missing doc})\\\)
Then substitute using Conditional Probability:
\(\displaystyle \Pr(\text{success})=\Pr(\text{sucess}|\text{mistake})\Pr(\text{mistake})+\Pr(\text{sucess}|\text{lied})\Pr(\text{lied})+\Pr(\text{sucess}|\text{missing doc})\Pr(\text{missing doc})\)

Hope you can finish the rest.
 
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