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).
$P(C)=P(\bigcap_{n=1}^\infty\bigcup_{k=n}^\infty A_k)=\lim_{n\to\infty}P(\bigcup_{k=n}^\infty A_k)\leq\lim_{n\to\infty}(p_n+p_{n+1}+\cdots)=0$.
#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 that should give you a WOW! feeling. Please make sure your supervisor shows you the beautiful solutions, if you have not found them yourself.
#16 Some students found this hard because they were not really sure what I was asking you to show. I have rewritten the question and posted a new version of Sheet 1. See if the new version on #16 is easier to understand. That may help your successors in IA 2015. Note that 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 and will help your successors.
#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).
$P(C)=P(\bigcap_{n=1}^\infty\bigcup_{k=n}^\infty A_k)=\lim_{n\to\infty}P(\bigcup_{k=n}^\infty A_k)\leq\lim_{n\to\infty}(p_n+p_{n+1}+\cdots)=0$.
#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 that should give you a WOW! feeling. Please make sure your supervisor shows you the beautiful solutions, if you have not found them yourself.
#16 Some students found this hard because they were not really sure what I was asking you to show. I have rewritten the question and posted a new version of Sheet 1. See if the new version on #16 is easier to understand. That may help your successors in IA 2015. Note that 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 and will help your successors.
