Wednesday, 11 February 2015

Examples Sheet 1

Most of you will have now done Examples Sheet 1. Let me make a few comments about some things I hope you learn from that sheet.

#7. (about infinite sequence of coin tosses and limsup of events) This was done in Section 4.4 of the notes. An even quicker solution can be done using the fact that $P$ is a continuous function (proved in Section 7.1 of notes).

$P(C)=P(\bigcap_{n=1}^\infty\bigcup_{k=n}^\infty A_k)=\lim_{n\to\infty}P(\bigcup_{k=n}^\infty A_k)\leq\lim_{n\to\infty}(p_n+p_{n+1}+\cdots)=0$.

This was part of the tripose question 2014, 2, II9.

#8. (sets none of which is a subset of the other) Think how this question would look if everything except the first and last sentences of its statement were removed. It would look hard! See how adding a layer of probability helps prove a theorem that seems to have nothing to do with probability. We saw this in Example 4.5 (Erdos's probabilistic method) and we will see this a couple more times in the course, next in Section 11.4 (Weierstrass approximation theorem).

#13 (Mary tosses more heads than John) and #18 (41 bags of red and blue balls). Both these questions have very beautiful solutions. Please make sure your supervisor shows you the beautiful solutions, if you have not found them yourself.

#16 This question relates to what I did in Lecture 10 about information entropy.

#17 (goat ate the cards!) I invented this question as a sort of joke - but I am pleased that it has turned out to be very thought provoking for many students. If the "penny has dropped" for you about this question then you are on your way to developing some good intuition in the field of probability.

Comments are welcome if you have any ideas for good questions on this part of the course.