Basic Engineering Probability and Statistics
Recitation 5: 23-26 September 2011
1. Roulette is one of the most popular casino games played across the world. In this problem we will
examine the probabilities associated with this “game of chance”.
The game consists of spinning a wheel containing slots for the numbers 1 to 36, and the numbers 0
and 00. Each of the numbers 1 to 36 is coloured either red or black (with exactly 18 red numbers
and 18 black numbers), and the other two are coloured green. A ball is tossed around the wheel and
eventually lands in one of the slots, selecting that number. It is assumed (and should be practically
the case if the wheel is properly manufactured) that the outcome of one spin has no impact on the
outcome of any other spin, i.e. that spins are independent, and that on each spin, every number is
equally likely to be selected.
Gamblers play by betting on the outcome of each spin, by placing chips on a layout representing the
possible outcomes (see last page). Bets can be made on individual numbers or on certain combinations,
such as strings of consecutive numbers, low or high numbers, or the colour of the number.
Bets on individual numbers pay 35 to 1, so if $1 is bet on, say, 12, and that number comes up, the
gambler keeps the $1 and wins an additional $35. Bets on combinations of numbers are paid in such a
way that you win the same betting on the combination as you would if you bet on each of the numbers
in the combination individually, with the same total bet amount. For example, a bet of $1 on
wins $17, which is the same amount you would win if you bet 50 cents on both 17 and 18. Similarly,
a bet on Red pays 1 to 1, which is the same payoﬀ you would see as if you bet individually on each of
the 18 red numbers.
(a) If you bet $1 on a combination bet consisting of
numbers, how much will you get paid if you
win (ﬁnd a formula in terms of
(b) Let the random variable
denote the outcome of one spin.
will take one of the values
. What is the probability
= 12)? What is the probability
any of the possible outcomes
? What (named) type of discrete distribution does this correspond