# Brainteasers

The following Brainteasers have been taken from " wilmott.com. They are all the type of questions you could easily face during a job interview. Some of these questions are simple calculation exercises, often probabilistic in nature reflecting the importance of understanding probability concepts, some have a 'trick' element to them, if you can spot the trick you can solve them, otherwise you will struggle. And some require lateral, out-of-the-box, thinking.

## The Questions

### Russian roulette

I have a revolver which holds up to six bullets. There are two bullets in the gun, in adjacent chambers. I am going to play Russian roulette (on my own!), I spin the barrel so that I don t know where the bullets are and then pull the trigger. Assuming that I don't shoot myself with this first attempt, am I now better off pulling the trigger a second time without spinning or spin the barrel first? (Thanks to pusher.)

### Matching birthdays

You are in a room full of people, and you ask them all when their birthday is. How many people must there be for there to be a greater than 50% chance that at least two will share the same birthday? (Thanks to bag head.)

At a cinema the manager announces that a free ticket will be given to the first person in the queue whose birthday is the same as someone in line who has already bought a ticket. You have the option of getting in line at any position. Assuming that you don t know anyone else s birthday, and that birthdays are uniformly distributed throughout a 365-day year, what position in line gives you the best chance of being the first duplicate birthday? (Thanks to amit7ul.)

### Biased coins

You have n biased coins with the kth coin having probability 1/(2k + 1) of coming up heads. What is the probability of getting an odd number of heads in total?

(Thanks to FV.)

When flipping an unbiased coin, how long do you have to wait on average before you get two heads in a row? And more generally, how long before n heads in a row.

(Thanks to MikeM.)

### Balls in a bag

Ten balls are put in a bag based on the result of the tosses of an unbiased coin. If the coin turns up heads, put in a black ball, if tails, put in a white ball. When the bag contains ten balls hand it to someone who hasn't seen the colours selected. Ask them to take out ten balls, one at a time with replacement. If all ten examined balls turn out to be white, what is the probability that all ten balls in the bag are white?

(Thanks to mikebell.)

### Sums of uniform random variables

The random variables X1, x2, x3, ... are independent and uniformly distributed over 0 to 1. We add up n of them until the sum exceeds 1. What is the expected value of n? (Thanks to balaji.)

### Minimum and maximum correlation

If X, Y and Z are three random variables such that X and Y have a correlation of 0.9, and Y and Z have correlation of 0.8, what are the minimum and maximum correlation that X and Z can have? (Thanks to jiantao.)

### Airforce One

One hundred people are in line to board Airforce One. There are exactly 100 seats on the plane. Each passenger has a ticket. Each ticket assigns the passenger to a specific seat. The passengers board the aircraft one at a time. GW is the first to board the plane. He cannot read, and does not know which seat is his, so he picks a seat at random and pretends that it is his proper seat.

The remaining passengers board the plane one at a time. If one of them finds their assigned seat empty, they will sit in it. If they find that their seat is already taken, they will pick a seat at random. This continues until everyone has boarded the plane and taken a seat.

What is the probability that the last person to board the plane sits in their proper seat? (Thanks to Wilbur.)