Reasoning With Mathematics
Lesson 9 - Making Bets and Decisions: The Statistics of Gambling
We've been discussing the way numerical patterns can help us to make decisions in a wide variety of
situations where human beings want to predict what will happen. Humans are so interested in such
prediciting that it is the basis one of the oldest forms of human play. When anthropologists try to
distinguish what makes us different from other animals, they argue about language, opposable thumbs,
weapons making, child rearing and a variety of other human characteristics or behaviors, as defining
characteristics of the species. Perhaps they should add guessing to their list, and guessing's close
cousin, gambling. We're going to spend some time studying gambling, or gaming as it is sometimes
called, to understand how patterns enable skilled gamblers to make better bets and enable all casinos,
lotteries, bingo games, pull tabs, racetracks, betting wires and punch cards to make money for the
people who run them. In the process we'll also learn some basic ideas about statistics and how they
help us to see patterns and to use those patterns to assess risk and make decision in much the same
way that Shirley did in figuring out whether or not she should take that risky winter drive to the cities.
Let's put Shirley on plane and send her of on a Vegas vacation, rescuing her from her winter wasteland. She braved the drive, got lucky and dodged the blizzard, got the job and has some extra cash. She wants to play and she wouldn't mind winning enough to pay all her expenses, and if she got really lucky again, a new four wheel drive truck to take a bite out of winter.
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If she wants to maximize her chances of winning, what game should she play when she gets to
Vegas?
Click on as many of these games as you want and then give an answer.
Poker <http://www.contrib.andrew.cmu.edu/org/gc00/reviews/pokerrules>
Blackjack <http://www.netcore.ca/~billk/vegas_casino.2.html>
Roulette <http://www.gamblingtimes.com/basroult.html>
Craps <http://www.gamblingtimes.com/bascraps.html>
Other <http://www.gambling.com/directory/directory.cfm?ID1=293>
Unsure <http://www.arizonaweb.com/gambling/gamble4.html>
{Ideally there would be a form here that allows students to select an alternative. Al l responses would bring students to <http://www.arizonaweb.com/gambling/gamble4.html> The Unsure response would simply bring them to the same site immediately }
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It's Those Tumbling Dice
Hallelujah, great odds!!!
The Tumbling Dice site announces that Craps is the best game for Shirley to play and proceeds to justify that claim by making a statistical argument that we are going to spend some time understanding. What this discussion demonstrates is a method that is at the heart of all reasoning with statistics and that can be extended to the most complex explorations of pattern that the human mind is capable of. In fact, the branch of mathematics that is called statistics really developed from the efforts of a eighteenth century mathematician, Thomas Bayes <http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Bayes.html>,
who was interested in finding "a method by which we might judge concerning the probability that an
event has to happen, in given circumstances, upon supposition that we know nothing concerning it but
that, under the same circumstances, it has happened a certain number of times, and failed a certain
other number of times." Today Bayes search for patterns is at the center of scientific study of everything
from computers to life and the brain. But let's start with a simple problem, a pair of dice and listing all
the possible outcomes we can get throwing that pair of dice.
Here they are.
Chart of Possible Outcomes - One Throw of a Pair of Dice
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
If we count the total number of potential outcomes (or multiply the number of faces on each di by each
other) we get 36 potential outcomes, one 2, two 3's, three 4's, four 5's, five 6's, six 7's, five 8's, four 9's,
three 10's, two 11's and one 12.
For any single throw of the dice then what are the chances we will get Or 2?
Think about our discussion of the binomial distributions. In order to determine the chances of making a
particular number of trips without encountering a blizzard we counted all the potential outcomes, given a
certain number of trips, and found we had one chance in four, eight, sixteen, etc. of avoiding the
blizzard. Here we have a similar situation, but the outcome on any particular trial (trip or throw of the
dice) is not the simple either/or outcome that the binomial pattern describes. For each one of Shirley's
trips, remember, there was a 50/50 chance, or one chance in two, that she would avoid the blizzard.
Each time we throw a pair of dice we have 36 possible outcomes.
How may ways can we get Or 2?
That's right, there is only one way to get 2. Our chart of 36 possible outcomes indicates exactly that.
So we have one chance in thirty-six of getting a 2 on any particular throw. Statisticians say the
probability of getting a 2 on a throw of the dice is 1/36 expressed as a fraction. In word it is the number
of different ways we can get a particular total divided by the total number of outcomes. Looking at the
chart we can see that the probability of each total is revealed by the pattern on the chart.
{The bold face numerals in these charts should be on the screen in colors (e.g. blue 3, red 4, etc.)}
For 3
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
two chances in thirty-six or a probability of 2/36.
For 4
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
three chances in thirty-six or a probability of 3/36.
For 5
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
four chances in thirty-six or a probability of 4/36.
By extension then, the probability of getting a 6 is 5/36;
a 7, 6/36.
And now, symmetrically, the table shows the probability of an
8 is 5/36;
9, 4/36;
10, 3/36;
11, 2/36 and
12, 1/36.
Now let's send Shirley back to the Craps table by way of
It's Those Tumbling Dice
Hallelujah, great odds!!!
<http://www.arizonaweb.com/gambling/gamble4.html>
According to the site the "odds" of winning at Craps are better because there are fewer possible
outcomes and better chances or probabilities of winning because the rules of the game specify
more winning outcomes than other games provide. Let's examine his explanation.
Each spot on the six-sided dice used in the game is worth one point. If you roll a seven or 11 on
the first roll, you win automatically. If you roll repeatedly "7"s you will double your money each
time, until a point is established. Out of all the combinations that could be rolled, there are only
two that add up to 11 points - if you roll a five first and a six second, or a six first and a five
second. Statistically, that's one eighteenth of the possibilities. However there are several
combinations for seven. They are 6,1; 1,6; 5,2; 2,5; 4,3; and 3,4. That's one-sixth of the
possibilities. Combined, you have approximately an 18 percent chance of winning on the first
roll. If the player rolls a 2, 3 or 12 on the first try, he/she automatically loses. There's only one
combination for 2 - 1,1, and only one for 12 - 6,6. A three shows up with 1,2 and 2,1.
Mathematically speaking, there's approximately a 12 percent chance of losing on the first roll.
If we look at the chart of outcomes again we can see the rationale for this view.
We win if we roll 7 or 11.
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
We have 8 chances in 36 of winning or a probability of 8/36 that we will win. If we actually divide 8 by 36
and multiply by 100, we get the percentage chance of approximately 22.22%.
Why, then, does our gambling advisor tell us that we have only an 18% chance of winning on the
first roll?
Before we can answer that question we'll need to define winning and recognize more about the ways in
which casino gambling hedges its bets to increase the "take" from gamblers. We'll also need to count
the number ways we can lose. We'll start there.
I lose if I roll 2, 3, or 12
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
I have four chances in thirty six or a probability of 4/36 of losing. That's an 11.11% chance of losing.
What happens if I don't role either a 7, 11, 2, or 3? I neither win nor lose. I have to roll again.
What is the probability of neither winning or losing?
I neither win nor lose on the first roll if I roll 4, 5, 6, 8, 9, or 10.
2 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 9
5 6 7 8 9 10
6 7 8 9 10 11
7 8 9 10 11 12
I have twenty-four chances in thirty-six or a probability of 24/36 of neither winning or losing. That's a
66.67% chance. Incidentally, the probability of an event occurring can be expressed either as a fraction
or as a decimal. So the probabilities for our first role of the dice are:
Winning 8/36 or .2222
Losing 4/36 or .1111
Roll Again 24/36 or .6667.
Suppose we assume I roll a 4 on my first throw. In order to win now, I must roll another 4 before I roll a
7 or an 11 which will cause me to lose. I now have three chances in thirty- six or a probability of .0833 of
rolling a matching 4 and wining on my next throw and a probability of .2222 of losing. The probability of
needing to roll again stays the same, .6667. Clearly my chances of losing have increased considerably.
While the situation differs slightly if I role a 5, 6, 8, or 9, my probability of losing .2222, is always greater
than my probability of winning. For a 5 or 9 the probability of winning is .1111; for 6 or 8, probability is
.1389 and 10 like 4 has a probability of .0833. Not only is this the case, but if I ask myself the how likely
is it that I will roll a 7 or 11 before I roll any one of the other numbers I discover that, like Shirley trying to
make consecutive trips to the cities without having her parade snowed on, my chances of losing keep
getting larger. In fact, as the number of times I roll to get my point increases, the binomial distribution,
the pattern that we used in evaluating Shirley's snow chances, becomes a good approximation for just
how bad my chances are. Flipping a coin 10 times trying to get heads each time, I'd have approximately
99 chances in 100 (probability = .99 or a 99% chance) that I would lose. Throwing dice I would have
approximately 92 chances in 100 (probability = .92 or a 92% chance) of losing. Those of you who would
like a more detailed explanation of how the binomial distribution is used in calculating probabilities
might want to visit Surfstat <http://frey.newcastle.edu.au/Stats/surfstat/surfstat.html> and read section
3-2-3 of Variation and Probability by Keith Dear. Dear's online statistics text is an excellent source for
understanding the mathematics behind basic statistical concepts. We will refer to it again as we
continue our discussion of
Though we have now gotten a clear idea of what the odds of rolling a winning combination are (or what
the odds of guessing correctly what number will be rolled are), we haven't really got a true picture of the
way the casino game of Craps operates. In fact, the casino operators set up rules for playing in their
games which enable players besides the dice roller to play (bet on outcomes), and that insure that no
matter how much various players win, the casino, or the "house," (as it's called), always wins. Let's visit
another website for an explanation. When you get to The Basics of Craps site,
<http://www.gamblingtimes.com/bascraps.html>
scroll down to the section titled:
Bettin' Them Bones
and read through the explanation of how the casino pays off on various bets. Here we have the heart of
how the casino makes money no matter who wins how much in any particular game and the way to
understand why our web "expert's" explanation doesn't quite match our probability calculations. We will
also discover that everything that you find on the world wide web is not necessarily reliable or accurate.
For the moment lets focus on only one statement from the Tumbling Dice site.
Our unknown author says :
Out of all the combinations that could be rolled, there are only two that add up to 11 points - if
you roll a five first and a six second, or a six first and a five second. Statistically, that's one
eighteenth of the possibilities. However there are several combinations for seven. They are 6,1;
1,6; 5,2; 2,5; 4,3; and 3,4. That's one-sixth of the possibilities. Combined, you have
approximately an 18 percent chance of winning on the first roll.
We previously wondered
Why does our gambling advisor tell us that we have only an 18% chance of winning on the first
roll?
The Basics of Craps site suggests one way that we can reconcile the apparent difference. The house
rules are set up not to affect any one roll of the dice, but to change the odds used in betting on the dice
so that over time, instead of breaking even, the house makes money by letting people bet on the
outcomes of games. The rule for rolling 7 or 11 on the first roll of a new game according to Basics of
Craps site is:
SEVEN: This one-roll bet pays odds of 4 to 1, correct odds are 5 to 1 with the difference giving
the house a 16.66% edge.
If the betting rule were designed to produce fair bets insuring that the Casino had no more chance of
winning than the player the payoff odds would be the correct odds, they would reflect accurately the
probability of throwing a 7 or 11. The house changes the odds to insure that it will make 1/6 more
money than winning its fair share of all wagers that a 7 or 11 will be rolled on the first throw in a turn or
pass would entitle the house to winning. It would wins such wagers four times out of five but pays off as
if it won only three times out of four. But Casino's and all other forms of gambling are not about fair
bets; they are about making money for the people who run the games.
In fact, if we do an exact calculation of the odds we can find out where that 18% chance of winning
came from. We simply need to use the ratios of odds to percentages to arrive at percentage chance we
have of winning money after the odds are adjusted, if we repeatedly bet on roling a 7 or 11 on the first
roll of a turn or pass.
True % chance of winning = 22.22 Odds 5 to 1
___________________________ ___________
=
Unknown % chance of winning Odds 4 to 1
Or
Unknown % chance of winning = 4/5 of 22.22%
= a 17.78% chance of winning
Or
approximately an 18% chance of winning.
In fact, on any one role of the dice on the first role of a pass the probability that we will role a 7 or 11
remains .2222 or 8/36, but we will be paid as if the probability were .1778. We will be paid less for
winning than we should be if we were playing by fair rules, that is rules based on the real probability of
outcomes. Look over the rest of the rules summarized in The Basics of Craps, Bettin' Them Bones
section and write a one page explanation of why and how all the Craps strategy pages advise bettors to:
play the line and the come, either pass or don't pass. These are the two best areas to bet,
offering the best possible odds to the player. If you're betting the pass line, always take
you full odds in back of your pass line bet. Some casinos offer double odds or higher; if
so, take advantage of this option. One last piece of advice: increase your bets on wins, do
not double up on losses.
If you have difficulty figuring just what this advice means visit several of the sites listed at The Gambling
Times Craps site <http://www.gamblingtimes.com> and you should be able to get a good idea of how to
bet and why this is the best bet.
As we go on in examining how we guess and gamble in all areas of our lives some of these same issues will come back to haunt and enlighten us about the ways we manage the risks of being human, of engaging in species specific behavior.