PA-4 / Lesson 2
Please do not forget to specify your probability model in each of your solutions! It takes 1-2 sentences, but it is a good exercise to not forget them. It is important.
Problem 1.
There are three boxes, each of which contains balls with numbers from 0 to 9. One ball is taken out of each box. What is the probability that three a) identical b) different balls are chosen?
Problem 2.
On each of 4 cards a distinct natural number is written. If two are picked at random, the probabilities that the sum is less than 9, equal to 9, and more than 9 are all equal. What numbers could be written on the cards?
Problem 3.
We flip a coin 1000 times. What is the probability of getting exactly 500 heads and 500 tails? Hint: it can be worked out using Stirling’s approximation.
Problem 4.
Both of the following problems could be solved using a similar idea:
a) A 5-digit number is picked at random. Find the probability that its digits are in strictly increasing order.
b) There are 12 books on a shelf, five of them are chose at random. Find the probability that no two of the chosen books are next to each other.
Problem 5.
Suppose a coin is tossed $n$ times, and let $p$ be the probability of heads. What is the probability that the total number of heads will be even?
There is a bizarre theorem, called Pólya's Random Walk Theorem, that points out an interesting difference between different dimensions. In its most general form, this theorem is about a $d-$dimensional space, but we will only look at the cases of 1D, 2D and 3D (which are the essence of the theorem anyway):
1D-case: Imagine a drunk person on a line, who each minute takes a step left or right, with equal probability, i.e $50\%$. Then with probability $1=100\%$ sooner or later he will come back to the starting point;
2D-case: Now, imagine a drunk person on a 2D plane, who each minute takes a step left, right, up or down with equal probability, i.e $25\%$. Then, just like in 1D case, with probability $1=100\%$ sooner or later he will come back to the starting point.
3D-case: Finally, imagine a drunk bird in 3D space, who each minute moves a fixed distance in one of the eights direction with equal probability, i.e $12.5%$. Then, unlike 1D or 2D, there is a chance that this bird will never ever come back. In fact, it can be shown that the probability of ever returning for this bird decreases to roughly 34%... Sad.
This result could be quickly summarised into the following slogan: "A drunk man will find his way home, but a drunk bird may get lost forever". There is a high-school friendly (but difficult-to-come-up-with) proof of it by the way, it relies on Stirlings Approximation as well as on the basics about generating functions (well, the very basics of them, so still something that a high-schooler can understand).