1. (10 points) There are 9 coins $c_{1}, c_{2}, \ldots, c_{9}$. Each coin has a different distribution: - The probability $p_{1}$ of coin $c_{1}: \quad p_{1} (H)=1 / 10 \quad p_{1}(T)=9 / 10$ - The probability $p_{2}$ of coin $c_{2}: \quad p_{2} (H)=2 / 10 \quad p_{2}(T)=8 / 10$ - The probability $p_{3}$ of coin $c_{3}: \quad p_{3}(H)=3 / 10 \quad p_{3}(T)=7 / 10$ - The probability $p_{4}$ of coin $c_{4}: \quad p_{4} (H)=4 / 10 \quad p_{4}(T)=6 / 10$ - The probability $p_{5}$ of coin $c_{5}: \quad p_{5} (H)=5 / 10 \quad p_{5}(T)=5 / 10$ - The probability $p_{6}$ of coin $c_{6}: \quad p_{6} (H)=6 / 10 \quad p_{6} (T)=4 / 10$ - The probability $p_{7}$ of coin $c_{7}: \quad p_{7}(H)=7 / 10 \quad p_{7}(T)=3 / 10$ - The probability $p_{8}$ of coin $c_{8}: \quad p_{8} (H)=8 / 10 \quad p_{8}(T)=2 / 10$ - The probability $p_{9}$ of coin $c_{9}: \quad p_{9} (H)=9 / 10 \quad p_{9}(T)=1 / 10$ (Note that $c_{5}$ is a fair coin.) Suppose your friend selects one of the coins uniformly at random. Your friend flips the coin 10 times and tells you that it came up $H 5$ times and $T 5$ times. Based on this information, what is the probability that your friend selected $c_{5} ?$ CS.VS. 1111