Thursday, 25 May 2017

How to win 1 M€ with 99% probability?

Play no limit roulette and bet 1 M€ on either black or red and always double your bet until you've made 1 M€ profit (then stop and never look back).

The joke is that you need to be able to go 127 M€ into debt to have 99% probability of winning 1 M€ as you need to be prepared to double in case you lose to win back your money and reach your goal. Your expectation value is always negative as you are likely to lose quite a lot when you do, but if you happen to have access to 127 M€, your chances of winning 1 M€ are approx. 99%.

A rational agent should never bet on negative expectation value, but given 99% rate of success, how many of us wouldn't? Besides of course banks and such to whom 127 M€ is nothing and only cumulative long term profits matter.

European roulette has slightly better odds than American since European has only one 0 instead of two (0 and 00). I wonder what's the limit in Monte Carlo?

If you have only 7 M€ limit (and start with 1 M€ bet), your chances of winning 1 M€ are still around 86%. If you have 100 k€ debt limit and start with 100 k€ bet, your chances of winning 1 M€ are 5.6%.

Beware the Barrenness of a Busy Life

clear all
MAXDEBT = 127;
for x = 1:1e6                       % test the strategy
    money = 0;                      % start with no money
    bet = 1;                        % starting bet
    while money < 1                 % play until you made 1 unit
        money = money - bet;        % bet
        if rand < 18/37             % roulette (18 red, 18 black and 0)
            money = money + 2*bet;  % if you win, you win double your bet
        end                         % otherwise lost the bet
        bet = 2*bet;                % double the bet after each round
        if money-bet<-MAXDEBT       % stop if at the limit
    mon(x) = money;

Ex = mean(mon)                      % ~ -0.2
MIN = min(mon)                      % -127
Pwin = sum(mon>0)/x                 % ~ 0.99

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