## Wednesday, February 25, 2015

### Unizor - Probability - Advanced Problems 1

Naive gambler decides to play roulette (36 red and black numbers from 1 to 36 to bet on and losing 0 and 00) against a casino, always betting on a color red (with a probability of winning 18/38 and the payoff on par with the bet, that is if you bet 1 bitcoin and win, you get your 1 bitcoin back and another from the house).
The strategy he decides to follow is to quit playing as soon as he wins the first time and, while he loses, to double his bet for each new spin of the wheel.
The first bet is 1 bitcoin. His capital allows him to lose no more than N times in a row, losing 1 bitcoin on the first spin, then 2 bitcoins on the second spin, then 4 bitcoins on the third spin etc. up to 2^(N−1) bitcoins on the Nth spin.

Problem A
What is the minimum amount of money a player needs, if he wants to be able to spin the wheel N times, while losing all the times?

Problem B
What are the elementary events we can define in this game? In other words, what are the mutually exclusive outcomes of this game?

Problem C
What are the probabilities of these elementary events?

Problem D
Prove that the sum of the probabilities of all mutually exclusive outcomes of this game equals to 1.

Problem E
What is the expectation of the winning (or losing) in this game? In other words, what is the average win (or loss) for this naive player?

Problem F
Is this strategy good for a player? Is it better to play it with more money?