Question

6.

A friend of yours thinks that he has devised a purely mathematical way of beating the standard European roulette

wheel, which has

37

pockets. His plan is to come to the roulette table with

100

units with which to bet. On each

spin, he places a one-unit, single-number bet. (Recall that winning such a bet returns

36

units to the winner.) He will

make exactly

100

bets, no matter how much he wins or loses. He claims that this strategy results in a greater than

50

%

chance that he will come out ahead at the end of

100

spins, so using this strategy repeatedly will make him a winner.

a)

Does he have a greater than

50

% chance of coming away with more money than he started with?

b)

If the answer to the previous question is yes, then does that make his strategy a winning one, meaning that it will

lead to a long-term increase in bankroll?

Answer 1

Europen Roulette has 37 pockets from 0,and from 1-36.

His plan is to come to the roulette table with 100 units with which to bet. On each spin, he places a one-unit, single-number bet. (Recall that winning such a bet returns 36 units to the winner.)

The straight bet, where one bets on a specific number and wins, with a payoff at 36-1, if that number comes up. This means that if a player bets $1 and wins, the player will get $37 back, the dollar he bet along with $36 more.

The probability of getting a specific number in one spin = 1/37

A) Does he have a greater than 50 % chance of coming away with more money than he started with?

So he can make 100 bets from $100 of $1 each if he wins at least three times which will amount to $37*3 = $111. He will return with more money than he started

P(b) = Probability of winning the bet = 1/37

n = number of trials(spins) = 100

r = number of times our bet wins = 3

P(E = Bet b appearing exactly r times in n spins] will be given by the binomial distribution

P(E at least 3 times) = 1 - P(E0) + P(E1)+P(E2) = 1- 0.0645+ 0.1741+0.2439 aprrox 0.51

So, yes he have a greater than 50 % chance of coming away with more money than he started with.

B) If the answer to the previous question is yes, then does that make his strategy a winning one, meaning that it will lead to a long-term increase in bankroll?

If we consider a straight bet of $1 of the above experiment and X to be the net winnings for the player, then X can take on either the value of 36 (if the player wins) or ?1 (if the player loses). If we denote the probability X takes on a certain value k by P(X = k),

So the expected loss is 2.7% for every $1 bet or we can understand it like for every $1 bet there is a chance of losing 2.7 cents

So, it will not lead to increase in bankroll.