There was a hit-and-run incident involving a taxi in a city in which 85% of the taxis are green and the remaining 15% are blue. There was a witness to the crime who says that the hit-and-run taxi was blue. Unfortunately this witness is only correct 80% of the time. What is the probability that it was indeed a blue car that hit our victim? (Thanks to orangeman44.)

Annual returns

Every day a trader either makes 50% with probability 0.6 or loses 50% with probability 0.4. What is the probability the trader will be ahead at the end of a year, 260 trading days? Over what number of days does the trader have the maximum probability of making money? (Thanks to Aaron.)

Dice game

You start with no money and play a game in which you throw a dice over and over again. For each throw, if 1 appears you win $1, if 2 appears you win $2, etc. but if 6 appears you lose all your money and the game ends. When is the optimal stopping time and what are your expected winnings?

(Thanks to ckc226.)

kgof berries

You have 100 kg of berries. Ninety-nine percent of the weight of berries is water. Time passes and some amount of water evaporates, so our berries are now 98% water. What is the weight of berries now?

Do this one in your head.

(Thanks to No Doubts.)

Urban planning

There are four towns positioned on the corners of a square. The towns are to be joined by a system of roads such that the total road length is minimized. What is the shape of the road? (Thanks to quantie.)

Closer to the edge or the centre?

You have a square and a random variable that picks a random point on the square with a uniform distribution. What is the probability that a randomly selected point is closer to the centre than to the edge?

(Thanks to OMD.)

Snowflake

Start with an equilateral triangle. Now stick on to the middle of each side equilateral triangles with side one third of the side of the original triangle. This gives you a Star of David, with six points. Now add on to the sides of the six triangles yet smaller triangles, with side one third of the 'parent' triangle and so on ad infinitum. What are the perimeter and area of the final snowflake?

(Thanks to Gerasimos.)

The doors

There are 100 closed doors in a corridor. The first person who walks along the corridor opens all of the doors. The second person changes the current state of every second door starting from the second door by opening closed doors and closing open doors. The third person who comes along changes the current state of every third door starting from the third door. This continues until the 100th person. At the end how many doors are closed and how many open? (Thanks to zilch.)

Two thirds of the average

Everyone in a group pays $1 to enter the following competition. Each person has to write down secretly on a piece of paper a number from 0 to 100 inclusive. Calculate the average of all of these numbers and then take two thirds. The winner, who gets all of the entrance fees, is the person who gets closest to this final number. The players know the rule for determining the winner, and they are not allowed to communicate with each other. What number should you submit?

(Thanks to knowtorious and the Financial Times.)

Ones and zeros

Show that any natural number has a multiple whose decimal representation only contains the digits 0 and 1. For example, if the number is 13, we get 13 x 77 = 1001.

(Thanks to idgregorio.)

Bookworm

There is a two-volume book set on a shelf, the volumes being side by side, first then second. The pages of each volume are two centimetres thick and each cover is two millimetres thick. A worm has nibbled the set, perpendicularly to the pages, from the first page of the first volume to the last page of the second one. What is the length of the path he has nibbled? (Thanks to Vito.)

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