This is taken from Examwise Vol 2 for the 2011 L1 cert. SS02. Q1. The probability of passing an eye test for people over 70 years old is 80%, and failing is 20%. Fr people under 70 years old the rates are 95% and 5% respectively. 15% of the people tested are over 70 years old. A report is found of someone failing the test. What is the probability that they are over 70 years old? Can someone work this out for me? The reason being is I think there is an error in the answer given in the book, and I want to check it.
What’s examwise?
Here’s a link to the book http://www.amazon.com/ExamWise®-Certification-Candidates-Chartered-Financial/dp/1590959620/ref=sr_1_2?s=books&ie=UTF8&qid=1307809665&sr=1-2
Is it any good? I hadn’t heard of it until now.
First you need to find probability of failing using conditional: Prob(Fail) = Prob(Fail|>=70) + Prob(Fail|<70) = 20%(15%) + 5%(85%) = 7.25% Then use Bayes: Prob(>=70|Fail) = [Prob(Fail|>=70)/Prob(Fail)] * Prob (>=70) = [20%/7.25%]*15% = 41.4%
yes, same answe as Andytrader, 41.38% for me. Use bayes formula. However, to determine the proba of fail (regarless of the age), you need the total probability rule which gives you P(fail) = 0.0725
Bayes theorem… YYUUCK!
andytrader Wrote: ------------------------------------------------------- > First you need to find probability of failing > using conditional: > > Prob(Fail) = Prob(Fail|>=70) + Prob(Fail|<70) > > = 20%(15%) + 5%(85%) = 7.25% > > Then use Bayes: > > Prob(>=70|Fail) = [Prob(Fail|>=70)/Prob(Fail)] * > Prob (>=70) > > = [20%/7.25%]*15% > > = 41.4% I thought we could get the same answer using tree diagram. If I try using tree diagram for this, I don’t get the same answer. Am I missing anything? Could someone demonstrate the same problem using tree diagram?
Cinderella Wrote: ------------------------------------------------------- > Bayes theorem… YYUUCK! Agreed! I didn’t even bother learning it for the exam.
using tree diagram: 41.3793%, I dont even know the formula really. Whats the right answer?
sgupta0827 Wrote: ------------------------------------------------------- > andytrader Wrote: > -------------------------------------------------- > ----- > > First you need to find probability of failing > > using conditional: > > > > Prob(Fail) = Prob(Fail|>=70) + Prob(Fail|<70) > > > > = 20%(15%) + 5%(85%) = 7.25% > > > > Then use Bayes: > > > > Prob(>=70|Fail) = [Prob(Fail|>=70)/Prob(Fail)] > * > > Prob (>=70) > > > > = [20%/7.25%]*15% > > > > = 41.4% > > I thought we could get the same answer using tree > diagram. If I try using tree diagram for this, I > don’t get the same answer. Am I missing anything? > Could someone demonstrate the same problem using > tree diagram? The tree diagram is tricky as it just shows you the path. It help you to know if you have to use only the total proba rule, the conditional proba, baye theorem or a combination of them. For this question, by drawing your tree (what I did), you realize that you need both total proba and bayes theorem. For the root of the tree, it is bayes theorem and for the leaves it is total proba.
I realized that this first step may not have been clear: Prob(Fail) = Prob(Fail|>=70)*Prob(>=70) + Prob(Fail|<70)*Prob(<70) = 20%(15%) + 5%(85%) = 7.25% Once you have probability of failing, then you can apply Bayes using known info.
yes you can just do this with a tree diagram. Say you have 100 people. 85 people are under 70, 15 are older than 70. Of the 85 people under 70, 80.75 pass, 4.25 fail Of the 15 people over 70, 12 pass, 3 fail. therefore the chance of someone being over 70 given that they failed is 3/7.25 = 41.4%
Kiakaha Wrote: ------------------------------------------------------- > yes you can just do this with a tree diagram. > > Say you have 100 people. > > 85 people are under 70, 15 are older than 70. > > Of the 85 people under 70, 80.75 pass, 4.25 fail > > Of the 15 people over 70, 12 pass, 3 fail. > > therefore the chance of someone being over 70 > given that they failed is 3/7.25 > = 41.4% Thanks Kia! Now I get it. I think I was confused about the language of the question.