9. Dominic randomly picks between two words MATHEMATICS and MEMES, each with an equal chance of popping up. From his chosen word, he then randomly draws one letter, with the probability of each letter popping up directly proportional to the number of times it occurs in the word. Given that Dominic drew an M, what is the probability that he, in fact, picked MEMES?
Answer: $\frac{11}{16}$
Solution: The probability of Dominic picking out the letter M from the word MATHEMATICS is $\frac{2}{11}$. On the other hand, the probability of him picking it from the word MEMES is $\frac{2}{5}$. This gives him, unconditionally, a probability of $\frac{1}{2}\left(\frac{2}{11}+\frac{2}{5}\right)=\frac{16}{55}$ of drawing the letter M. Then, the probability of him drawing the word MEMES, then the letter M from it, is $\frac{1}{2} \cdot \frac{2}{5}=\frac{1}{5}$. Hence, by Bayes' theorem, the probability that the word Dominic drew is MEMES is $\frac{\frac{1}{5}}{\frac{16}{55}}=\frac{11}{16}$.