**Exercise 1.14**:

Let be the event that we are equally talented, be the event that I am slightly better, be the event that he is slightly better, be the event that the game ends up in one win for me and three wins for him.

We have:

Apply Bayes’ Law:

Therefore in my posterior model, I should believe that my opponent is slightly better than I am with probability .

**Exercise 1.15**:

Let A be the event that their sum is divisible by 6.

The principle of deferred decision yields .

**Exercise 1.16**:

Let be the event that there are exactly of three dice have the same number on the -th roll, B be the event that the player wins the game.

**(a)**

**(b)**

**(c)**

**(d)**

In order to calculate we calculate two probabilities and

**Exercise 1.20**: Show that if are mutually independent, then so are .

We only prove for .

Since , and are independent.

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Posted by Man on June 30, 2017 at 6:06 pm

How do you get (c) of Exercise 1.16? Thanks.