Problem 1.

a) Suppose we toss a fair coin $2n$ times. Prove that the probability of equal number of heads and tails is approximately $ \frac{1}{\sqrt{\pi n}}$. For $n = 13$ this is around 0.156 (while the exact answer is around 0.155);
b) Find the probability of extracting 26 cards from a shuffled deck and obtaining 13 red and 13 black (once again, you can obtain a very decent approximation for it). You should get an answer larger than the answer from the previous part. Could you explain why it happens?

Problem 2.

Perla throws a fair coin 11 times, and Jason throws a fair coin 10 times. What is the probability Perla gets more heads than Jason?

Problem 3.

A number between 1 and $10^{10}$ is picked at random. What is the probability that all of the digits of this number are different and that it has both: one even and one odd digit?

Problem 4.

There is a convex $2k+1$-gon. A pair of its diagonals is chosen at random. What is the probability that the two chosen diagonals intersect inside the polygon?

Problem 5.

Suppose we toss a fair coin $2n$ times. We say that an equalization occurs at the $2k$-th toss if there have been $k$ heads and $k$ tails. Here are two things that might happen when we toss a coin $2n$ times:
1) No equalization ever takes place (except at the start), or
2) An equalization takes place at the end (exactly $n$ heads and $n$ tails).
Which do you think is more likely?

Problem 6.

a) There is a drunken person standing one step away from the lake (which is to his left) and just 10 steps away from the house (which is to his right). He is about to take exactly 100 steps (all of equal size), each being right or left with equal probability. What is probability that he will end up home on his 100-th step without ever ending up in a lake during his journey (he might go further to the right of the house)?
b) Continuing with ... let's call it more officially "random walking", remember the "what up story" from the Problem Set? (yeah, that is how that block of text is called: a "what-up story"). The whole result is rather difficult to prove, but try solving its "1D case".