## Three Probability Problems44

We toss a biased (an unfair) coin 10 times. The coin has a probability of tossing heads p. Calculate the probability that there are 5 heads in the first 8 tosses and 3 heads in the last 5 tosses (both of them must happen), in terms of p. [Solved by HardCarry]

Suppose there is a crazy professor who grades some submissions with marks {1, 2, 3, 4, 5, 6} and he does it totally randomly. How many times is the mean value of submissions, in which you will have every mark at least once? [Solved by HardCarry]

Suppose there are 2n persons who form n couples. Suppose that after many years, the probability of a person being alive is p, and it is equally likely for all persons. On condition that after many years, m people are alive, find the mean value of couples who have both persons alive. (on terms of m and n)

okay

In a class I taught at Berkeley, I did an experiment where I wrote a simple little program that would let people type either “f” or “d” and would predict which key they were going to push next. It’s actually very easy to write a program that will make the right prediction about 70% of the time. Most people don’t really know how to type randomly. They’ll have too many alternations and so on. There will be all sorts of patterns, so you just have to build some sort of probabilistic model. Even a very crude one will do well. I couldn’t even beat my own program, knowing exactly how it worked. I challenged people to try this and the program was getting between 70% and 80% prediction rates. Then, we found one student that the program predicted exactly 50% of the time. We asked him what his secret was and he responded that he “just used his free will.”

- Scott Aaronson