An urn has $100$ normal dice, plus $75$ dice whose face numbers are $2, 2, 4, 4, 6, 6,$ and plus $25$ dice whose face numbers are $1, 1, 3, 3, 5, 5$. One die was chosen at random. If I just rolled one $6$ with this die, what is the probability that I chose a normal die?
I have no clue how to go about this problem. Your help would be really appreciated.
You could argue completely with Bayes' theorem, but you can also observe that uniformly picking a die and rolling it once amounts to uniformly picking one of the $ 1200$ faces. You picked a six (out of $250$ sixes). There are $150$ sixes on "even" dice and $100$ sixes on normal dice (plus none on "odd" dice). Hence the probability that your particular six is on a normal die is $\frac{100}{250}$.