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)
You're watching the news and there's two weathermen on air currently. The first weatherman is 70% accurate, the second weatherman is 40% accurate. They both predict that it will rain today. What is the probability that is actually rains?
This isn't probability so much as thinking way too much outside the box, but for problem 3, based on how one infers the connotation of "many" years, it could easily 100 years, or 500 years, or many more than that. If one also assumes that these are average humans with a life expectancy of 80 years or so, then the answer is clearly 0, because they would all be dead.
