Engineering ENGRD2700
Basic Probability and Statistics
Fall 2011
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 {17, 18}
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 payo 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 n numbers, how much will you get paid if you
win (nd a formula in terms of n)?
(b) Let the random variable X denote the outcome of one spin. X will take one of the values
{1, 2, . . . , 36, 0, 00}. What is the probability P (X = 12)? What is the probability P (X = x) for
any of the possible outcomes x? What (named) type of discrete distribution does this correspond
to?
(c) Let W denote the random amount won if a bet of $1 is placed on the number 12 (losses are
considered negative winnings). Write W explicitly as a function of X . What is the expected
winning E (W )? What is the variance Var(W )?
(d) Let W be the amount won if $1 is placed on the combination {25,26,27}. What is E (W )? What
is the expected winning from betting $1 on any combination of n numbers out of {1, 2, . . . , 36}?
(e) What is the expected winning if a player bets $2 on {10, 11, 12}, $5 on {31, 32}, $3 on 20 and $10
on Red, all on the same spin? We can think of the winning as the random variable W given by
W = 2W1 + 5W2 + 3W3 + 10W4 , where the Wi are the winnings from each of the 4 bets.
(f) Recall the winning amount W from part (d). What is its variance? What is the variance of the
winning from betting $1 on Red?
(g) One possible combination is {0, 00, 1, 2, 3}. This bet pays 6 to 1 and is considered the worst bet
in the game. Lets see why. What should the payo be under the formula from part (a)? Note
that the actual payo does not agree with the formula, because payos are required to be integervalued and the formula does not give an integer in this case. (In fact, this is the only bet which
does not follow the formula.) What is the expected winning from betting $1 on this combination?
(h) We have found that the winning expected on all bets is negative, i.e. that the house expects to
win a small proportion from each bet placed. This means that the game is designed so that in
the long run the casino will win more than it pays out. This is the house advantage. How
is this advantage built into the game? Suppose the possible outcomes were only the numbers 1
through 36, without the 0 and 00. What would be the expected winning from betting $1 on the
number 12, or any number x? What then would be the expected winning from betting $1 on a
combination of n numbers? This means that in this case, roulette would be a fair game. How
then does the inclusion of the numbers 0 and 00 create the house advantage?
(i) Let X1 be the outcome of the rst spin, X2 the outcome of the second spin, and so on. Recall
that we are assuming that the outcomes of dierent spins are independent. What is P (X1 is red)?
What is P (X2 is red)? What then is P (X2 is red | X1 is red)? Similarly what are
P (X9 is red | X1 is red, X2 is red, . . . , X8 is red) and
P (X9 is black | X1 is red, X2 is red, . . . , X8 is red)? It is a common conception that after a run of
reds, we are due for a black, so black would be more likely on the next spin than it would be
normally. What can you say about this reasoning? Would you use this idea to guide your betting
strategy?
(j) Extra for Experts. Many people believe that it is possible to beat the odds and win more
than the (negative) expected winning most of the time by using appropriately chosen sequence
of bets. This is known as a betting system. One of the more well-known ones is the Martingale
system, which is supposed to guarantee a win. The system consists of the following: bet $1 on
Red. If you win, pocket the winning of $1 and start over, and if you lose, then bet $2 on Red.
Continue in this way, betting twice as much each time if you lost just before, and starting over if
you won. Under this system, you are (ideally) guaranteed to win $1 eventually.
How much will you win if you win on the second bet? How about the third bet? How about the
n-th bet? Since a red number is sure to come up eventually, you should be sure to win. The idea
is that by increasing the bet size, you recover previous losses and make a small prot on top of
that with one win.
This reasoning is correct, but there are practical problems in its implementation. If you keep
losing, your bets increase exponentially fast. If you start with a bankroll of $100, how many losses
would it take before you can no longer double your bet (so you wouldnt be able to recover all
your losses with one win)? How about if you start with $1000? What is the probability that this
happens (compute the probability separately for each cases)? Another issue is that many casinos
have a house limit, i.e. a maximum bet amount which applies to each spin. In light of this, would
you consider using this betting system?
2. A certain carmaker produces one model of car out of two factories. One of the factories has a conscientious manager who makes sure his workers are doing their jobs, whereas the manager of the other
factory spends most of his day in his oce surng the internet, and harldy ever checks up on the quality
of his workers labour. The distribution of the limetime in years of this particular car model has been
established and is given in the spreadsheet cars.xls. (It would seem reasonable that the shorter-lived
cars tend to be the ones produced in the second factory.)
Graph the pmf, and compute and graph the cdf. Compute the mean and variance of this distribution.