Probability

Brain teasers and puzzles involving randomness, probability, and statistics

Difficulty - Hard, Probability

Vacant Room Probability

Your workplace has a phone room for employees to quietly make personal calls. The room has no windows, just a sign that can be switched from “Vacant” to “Occupied”. However, employees differ in how consistently they use the sign:

1/2 of them always switch to “Occupied” when they enter and “Vacant” when they exit.
1/4 of them ignore the sign altogether – the sign will always read the same before, during, and after their visit.
1/4 of them always switching to “Occupied” when they enter, but always forget to switch back to “Vacant” when they exit.

If the room is actually occupied exactly 1/2 of the time, what is the probability the room is actually vacant when the sign reads “Vacant”?

Difficulty - Easy, Logic Puzzles, Probability

Random Turns

You are in a city with streets on a perfect grid – every street is north-south or east-west. You are are driving north, and decide to randomly turn left or right with equal probability at the next 10 intersections.

After these 10 random turns, what is the probability you are still driving north?

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Difficulty - Medium, Interview Brain Teasers, Probability

Tale of Two Trains

There are two trains that run between two cities. The trains are identical and run on identical routes, so passengers have no preference between the two and would take whichever train that pulls into the station. The trains run at the same frequency: exactly once an hour.

You often travel between the two cities on a whim, and when you do so, you show up at the station at a completely random time. Yet after many trips over the years, you notice that you have taken one of the trains three times as often as the other. Is this just really bad/good luck, or is there another likely explanation?

Difficulty - Hard, Interview Brain Teasers, Probability

Wrong Seat Probability

There is a fully-booked flight with 100 seats. The first person decides to ignore the seat assignments, and sits in a random seat. Each subsequent person sits in their assigned seats if available, or sits in a random unoccupied seat if not.

What is the probability that the last person happens to find their assigned seat unoccupied?

Difficulty - Medium, Interview Brain Teasers, Probability

Chance of Rain

You are driving to Seattle to meet some friends, and want to know whether you should bring an umbrella. You call up 3 of your friends who live there and independently ask them if it’s raining. Your friends like to mess with you, so each of them has a 1/3 chance of lying. If all 3 friends tell you it is raining, what is the probability it is actually raining there?

This question was asked in an actual Facebook data scientist/data analytics interview.

Preparing for a brain teaser interview? Check out our ultimate guide to brain teaser interviews.

Difficulty - Easy, Interview Brain Teasers, Probability

Game the Coin Flip Game

A small company is losing money and the boss is looking for ways to cut costs. The boss is a foolish gambler, so he offers two employees the choice of either taking flat 10% pay cut, or playing the following coin flip game for each paycheck:

The employees each flip a fair coin – 50% heads and 50% tails; the boss will ensure they are not pulling any tricks.
They can see their own outcome but not the outcome of the other employee’s coin flip. They must then guess the outcome of the other employee’s coin flip.
If at least one of them guesses correctly, they get their full paycheck.
If both of them guess incorrectly, they get nothing.

The two employees take some time to work out a strategy, and then confidently accept the offer to play the coin flip game.

What strategy did they come up with that made them confident the coin flip game will give them more money?