# 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.)

### Another one about birthdays

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.)

### Two heads

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.)