2n points on circle

 Read this puzzle somewhere. 2n points are distributed along a circle in equidistant manner. 2 points are chosen at random and a line drawn between them. 2 points are chosen again randomly from the rest of 2n-2 points and line is drawn between them. What is the probability these lines intersect (inside the circle).

Another way to frame the same question is to randomly select 2 points on the circle draw a line and then randomly select another 2. find probability on intersection.

 

2nd part: Given n chords chosen at random on a circle. What is the expected number of chords that will intersect.

Some similar puzleson inclusion-exclusion

Inclusion exclusion is a common technique for lost of combinatorics puzzles. Here are two I recently encountered:

0. At the banquet of a large conference, n mathematicians hang their coats on the coat rack as they enter. At the end of the night they leave in a drunken stupor, each one randomly putting on a coat without checking that it’s their own. Show that in the limit as n → ∞, the probability that none of the mathematicians staggers home in their own coat approaches 1/e. 

1. 6 persons are standing in a line what is the probability that no three consecutive people are in increasing order of their heightsalso see: https://artofproblemsolving.com/wiki/index.php/Principle_of_Inclusion-Exclusion

2. 6 babies are born in a hospital on either monday, tue, wed, thu. What is the probability that there was no day when no baby was born?

3. 10 people numbered uniquely in 1-10 are uniformly randomly given 10 tickets uniquely numbered b/w [1,10]. What is the probability that none of them get the ticket with same number as the number assigned to them. 

Probability $n$ uniform random points lie on a semicircle

Nice puzzle I read on Saurabh Joshi's blog. What is the probability $n$ uniform random points will lie on a semicircle.

Expected number of tosses for consecutive heads

Geometric distribution with parameter $p$ is the number of tosses of biased coint to get first head. The distributions is given by $P(X=k) = (1-p)^{k-1}p$. The expected number of tosses to get first head is $\frac{1}{p}$. Now, the question is to find the expected number of tosses to get the first HH pattern.

Puzzle #8: Three dart puzzle

You are throwing darts at a dart board, aiming at the center. The second dart hit the board farther from the center than the first. What is the probability the third dart will also hit the board farther from the center than the first? Assume that all the throws are independent.

Puzzle #6: Probability of centre of square inside circle

Select two points uniformly randomly inside a square. What is the probability that the center of the square will lie inside the circle drawn with these two points as ends of diameter?

Source: Rishab Vaid

Puzzle #4: Stop the roll

Problem: You are in a dicey situation. You friend gave you a dice and asked you to keep rolling till you get a sum of 100 or more. Now,you have to tell the most probable number at which you are going to stop.

Solution: You will stop at or before 105. Now, trick is to think backwards. You can get 105, only if you ever reach 99. Similarly, you can get 104, only from 99 or 98. You can get to 100, from maximum previous sums i.e. 94, 95, 96, 97, 98 and 99. Therefore, 100 is the most probable stopping point.  P(105) = P(105|99).P(99) = P(99)/6 . Similiarly, P(104) = (P(99)+P(98))/6, ...P(100)  = (P(94)+...+P(99))/6

Source: Rishab