(a)A gambler has a fair coin and a two-headed
coin in his pocket. He selects one of the coins
at random; when he flips it, it shows heads.
What is the probability that it is the fair coin?
(b) Suppose that he flips the same coin a second
time and, again, it shows heads. Now what is
the probability that it is the fair coin?
answer to (a) is 1/3 which you need for (b),
the answer to (b) is
I learned the basics of Bayes, but I don't understand what it means to have $O_1$ and $O_2$
Problem (c))
Suppose that he fluids the same coin a third time and it shows tails. What's the probability that it is the fair coin?
How do we solve this?
Best Answer
It appears that $O_1$ and $O_2$ are first outcome is heads and second outcome is heads. $P(F|O_1)$ is the probability that the coin is fair given that it comes up heads the first time is $1/3$, and the probability that it comes up heads the second time given that it is the fair coin and came up heads the first time is $1/2$. You can check that the numbers in the denominator are also consistent with this interpretation of $O_1$ and $O_2$.