(a) What is the difference between P(A|B) and of illustrative examples. (BIA) ?

Question

(a) What is the difference between P(A|B) and of illustrative examples. (BIA) ? Please explain by giving a couple (b) If P(B) = P(A), does it follow that P(AB) = P(BIA)? Please explain with simple mathematical reasoning (c) Suppose you want to know what your chances are to succeed at X, where x is something that matters to you and you want to do (say, graduate, make one billion dollars, write a novel, win the nobel prize, whatever). Suppose you learn from a very reliable source that those who succeed at X are very often of type Y, that is, P(a person is of type Yla person succeeds at X) is very high. You are a person of type Y (say, introvert or extrovert or shy or whatever). What is your reaction? Could you then be reassured that you are very likely to succeed at X since you are of type Y? Please explain and show me the logic of your argument (d) You are deciding whether to go to school A or B. They cost the same, are equally prestigious and you have been accepted to both. You care mostly about a highly paying job and it turns out that most of those who have highly paying jobs in your town studied at school A. That is, P(person studied at school Alperson has highly paying is higher than P(person studied at school Blperson has highly paying job). Is this a good reason to pick school A over B? Please explain and show me the logic of your argument (e) Luveko and Eresia are happy living together. They do a regular check-up one day and it turns out they both test positive for a rare disease which could cause serious health complications as they get older. They both start to get worried, but they try to remain calm and seek more information. Here is what they find out. The test is 99% reliable, but the disease is quite rare. It affects overall only 1% of the population. However, as it turns out, the disease affects males more frequently than females, and in particular, it affects males with blue eyes forty times as often as females with brown eyes. Luveko is a male with blue eyes, while Eresia is a female with brown eyes. We know the disease in question affects 20% of males with blues eyes Should Luveko and Eresia still be worried? How likely are they to have the diseas given that they tested positive? Please answer this question using Bayes' theorem an the method of counting cases. Make sure you show me the logic of your argument NB: What is most important here is that you show me your reasoning as clearly as you can.Y nust demonstrate you fully understand how to use conditional probabilities and Bayes theore tively. Do not be too worried about the numerical calculations. Of course, try to get yo alculations right, but the most important thing is that you get the logic of the argument right

Explanation / Answer

Answer to part a)

P(A | B) is conditional probability of occurence of A when B has already occured

P(B | A) is the conditional probability of occuance of B when A has already occured

.

P(A |B) = P(A and B) / P(B) , whereas P(B |A) = P(A and B) / P(A) , so we cna see that the denominator of these two formulae is differen

thus mathematically the two terms are different

.

part b)
Yes if P(A) and P(B) are equal then deifnitely P(A |B) = P(B |A)
P(A |B) = P(A and B) / P(B)

P(B | A) = P(A and B) / P(A)

Since P(A) = P(B) , thus it follows that P(A | B) = P(B | A)

.

Part c)

Lets say x is clearing SAT

Y is being studious

and I am studious

it is given that P(Studious | Cleared SAT) that is probability of a student being studious given he has cleared SAT is higher

this means more of the studious students are the one's who clear the SAT

thus I need to be happy since I have higher chance of clearing X = SAT

So if Probability of occurance of Y , when X has already occured is higher, this implies that if I have the Y quality , I have more chances of clearing X

.

Answer to part d)

given:

P(A | highest paying job) > P(b | highest paying job)

Using the conditional formula we get

P(A | Highest paying job) = P(A and Highest paying job) / P(Highest paying job)

P(B | Highest paying job) = P(B and highest paying job) / P( Highest paying job)

ON plugging the formulae in the first formula we get

P(A and highest paying job) / P(Highest paying job) > P(B and highest paying job) / P(highest paying job)
P(A and highest paying job) > P(B and highest paying job)

This means that there are more people from A to get highly paid as compared to people from B

thus it is definitely a valid reason to believe that school A results in highest paying jobs

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