Feb 25, 2011

Coin Toss Bankruptcy

Source: Mailed to me by Sudeep Kamath (EECS PhD Student, UC Berkeley, EE IITB 2008 Alumnus)

Problem: Three people start with integer amounts a,b and c. In each round, each one tosses a fair coin. If not all faces are the same, the person with the different face gets a rupee from each of the other two. If all faces are the same, no money is exchanged. This process is repeated till one of them gets bankrupt. What is the expected number of rounds till the game ends?

Related Problems:

Update (15/03/2011):
Hint: Given away by Sudeep. (* Define a martingale of the form
Y_n=A_n*B_n*C_n + some other term (where A_n,B_n,C_n are the fortunes
of the three players at time n). *)
Solution: Posted by chera (Gaurav Sinha, IITK 1996 Graduate, Indian Revenue Service) in comments!

Derangement - Complete FAIL!

Source: Interview at one of the quant firms

As posted in problems this and this, this is an extension to the derangement problem.
There are n men, n hats, one hat belonging to each person. A random permutation of hats is picked by the men. What is the probability that no person gets the correct hat?

Update (March 6, 2011)
Solution: Posted by Vipul Verma (Engineer at Portware, IIT KGP and University of South California Alumnus) in comments!

Feb 6, 2011

Equal Heads and Tail

Source: Posted by chera (Gaurav Sinha, IITK 1996 Graduate, Indian Revenue Service) in comments on Consecutive Heads Problem

Suppose you have a fair coin and you toss it until you have got equal number of heads and tails. What is the expected number of tosses? Note that probability that the game stops in odd number of tosses is 0. The probability that the game stops in 2 tosses = 0.5

Different solutions posted by Kalyan Parhi (EE IITB Alumnus), Abhash, Siva, Gaurav Sinha (CSE IITK 1996 Alumnus, Indian Revenue Service), Dinesh Krithivasan (IITM Alumnus, Phd University of Michigan, Senior Qualcomm Engineer) in comments!

Fraction Brainteaser

Source: Sent to me by Gaurav Sinha Problem: Siddhant writes a Maths test and correctly answers 5 out of 6 Arithmetic questions and 20...