Bayes Formula

Can somebody try and explain this to me, I don’t know why what I read in the book doesn’t make sense to me… I mean when do you use it and how? Thank You

You use Bayes theorem when someone gives you P(X given Y) but you really want P(Y given X). The archetypal example is that I have two coins in my pocket: Coin A: Fair coin Coin B: Coin with two heads I pull one coin out of my pocket, flip it, and it lands heads. What is the probability I pulled coin B out of my pocket? So you know stuff like P(H | flip A), P(H | flip B) but you want P(B | H). You can reason this one out by saying that there are three heads in my pocket and if you see one then it is either the head on A, head 1 on B, or head 2 on B. Since those are equally likely P(B) = 2/3 Using Bayes theorem, P(B | H) = P(H | B)*P(B)/(P(H | B)*P(B) + P(H | A)*P(A)) =1*1/2/(1*1/2 + 1/2*1/2) = 1/2 / 3/4 = 2/3 as advertised. In the real world, you don’t use Bayes theorem all that much. What you do use are Bayesian statistics which start with Bayes theorem calculations. The idea is that if you ask me to estimate the probability that a die comes up with a 6 on a roll, I can be stupid and say that I have no idea so I just need to roll the die n times, count the # of 6’s and divide by n. Classical statistics can then give you standard error estimates. On the other hand, a Bayesian would pick up the die, examine it with some calipers, come up with some uncertainty measures surrounding the calipers, and roll that into a distribution for his guess at the probability of a 6. He would then roll the die and use the die rolls to modify his prior guess. The Bayesian will always have a better guess with lower standard error than the silly frequentist above. Who would make a better securities analyst, the guy with the calipers or the guy who just shrugs and rolls the die? Edit: Typos and I forgot mynew signature again. ----

Thank you for the answer!!