To the question "Are you an engineer?", Jack answered "Yes".
Would you still argue that the probability is 30% that he is an engineer? A lawyer can claim to be an engineer, after all. However I think it is clear that if we actually did the experiment, it would be much more likely than 30% that Jack is an engineer. A way of testing what you really believe the probability to be is this: I bet you a dollar that Jack is an engineer. If you wouldn't, that means that you really believe the probability to be larger than 50%.
This is because the probability that he answers yes to the question is much higher when he is in fact an engineer than when he is a lawyer. Bayes' law says:
P(E|Y) = P(E) * P(Y|E)/P(Y)
You should read P(A|B) as "the probability that A is true given that B is true". In this case E = "a person is an engineer" and Y = "a person answers yes to the question 'are you an engineer?'". As you can see the original P(E) = 30% gets multiplied by P(Y|E)/P(Y) given the information that the person answered yes. The probability that a person answers yes given that he is an engineer is higher than the general probability that a person answers yes. So P(Y|E)/P(Y) > 1. So P(E|Y) > 30%.
This same law applies to other characteristics, for example Y = "person likes mathematics".
To the question "Are you an engineer?", Jack answered "Yes".
Would you still argue that the probability is 30% that he is an engineer? A lawyer can claim to be an engineer, after all. However I think it is clear that if we actually did the experiment, it would be much more likely than 30% that Jack is an engineer. A way of testing what you really believe the probability to be is this: I bet you a dollar that Jack is an engineer. If you wouldn't, that means that you really believe the probability to be larger than 50%.
This is because the probability that he answers yes to the question is much higher when he is in fact an engineer than when he is a lawyer. Bayes' law says:
You should read P(A|B) as "the probability that A is true given that B is true". In this case E = "a person is an engineer" and Y = "a person answers yes to the question 'are you an engineer?'". As you can see the original P(E) = 30% gets multiplied by P(Y|E)/P(Y) given the information that the person answered yes. The probability that a person answers yes given that he is an engineer is higher than the general probability that a person answers yes. So P(Y|E)/P(Y) > 1. So P(E|Y) > 30%.This same law applies to other characteristics, for example Y = "person likes mathematics".