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Three Probability Problems

Elfen
over 7 years

  1. 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.

  2. 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?

  3. 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)

Harkinian
over 7 years
https://pastebin.com/4rMUdnJg

revisions aaaaaaaaaaaaaa
Elfen
over 7 years
Edit: For the second problem, in the example of my pastebin, I gave you an outcome of 6 elements.

Another outcome could be {B+, C+, B+, A-, B, A-, B} which has 7 elements, and only 4 out of them are different numbers.

An outcome doesn't necessarily have to have 6 elements, it will (probably) have (much) more. Imagine that the outcome stops when you have all 6 grades at least once.
Elfen
over 7 years
First of all I want to clarify that it's not my homework and that I have solved the problems but I'm posting them if someone wants to try them.

The answers to Harkinian's submissions are here:

https://pastebin.com/06CaQswj
Harkinian
over 7 years
Here you are you lazy git

https://pastebin.com/DHepnZAB
deletedover 7 years
do your own homework