Xavier and Yvon have staked $10 each on a coin-tossing game. Each player tosses the coin in
turn. If it lands heads up, the player tossing the coin gets a point; if not, the other player gets a
point. The first player to get three points wins the $20. Now suppose the game has to be called
off when Xavier has 2 points, Yvon has 1 point, and Xavier is about to toss the coin. What is a
fair way to divide the $20?
The Beginning of Probability
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Archeologists have discovered dice that date back to thousands of years B.C.
Modern dice games didn’t become popular until the middle ages.
In 1654, Antoine Gombaud, the Chevalier de Mere, a wealthy French nobleman who
gambled for a hobby, approached Blaise Pascal with a mathematical problem(s) relating
to dice games.
Pascal began communicating with Fermat on these problems as well. These interactions
are what truly began the field of probability.
There is disagreement as to what problem actually started the conversations between de
Mere and Pascal, but it is clear that they discussed many different problems related to
probability.
One of these problems is today called the “Problem of Points” and can be stated as: Two
equally skilled players are interrupted while playing a game of chance for a certain
amount of money. Given the score of the game at that point, how should the stakes be
divided?
The Problem of Points
The problem I posed to you at the beginning of class is a simple version of this problem.
Xavier and Yvon are in the middle of a game of chance. The game must be interrupted. How
should the stakes be divided?
Pascal’s solution would have been as follows:
A fair coin is equally likely to turn up heads or tails. Thus, if each player had two points, each
would be equally likely to win the game on the next toss, so it would be fair for each player to
get $10, half of the staked amount at that stage. In this case, Xavier has 2 points and Yvon has 1.
If Xavier tosses the coin and wins, he has 3 points and hence gets the $20. If Xavier loses, then
each player has 2 points and hence each is entitled to $10. So Xavier is guaranteed at least $10
at this stage. Since it is equally likely that Xavier would win or lose on this toss, the other $10
should be split equally between the players. Therefore, Xavier should get $15 and Yvon should
get $5.
Using this strategy, determine how much Xavier and Yvon should get if they had been playing
the game, but had to stop the game after Xavier had 2 points and Yvon had no points.
Fermat’s solution to this problem would have been as follows:
Fermat computed each player’s probability of winning. In the original case where Xavier had
two points and Yvon had 1 point, Fermat would have determined:
There are two possibilities for the next throw; either Xavier wins or they are tied 2 to 2. If they
are tied, then one more throw needs to occur. In the latter case, they each have a 50-50 chance of
winning. So Xavier has ½ + ¼ = ¾ chance of winning but Yvon only has ¼ chance of winning.
Therefore, Xavier should get $15 (¾ of $20) and Yvon should get $5 (¼ of $20).
Using this strategy, determine how much Xavier and Yvon should get if they had been playing
the game, but had to stop the game after Xavier had 2 points and Yvon had no points.
The other of “the first problem in probability”
The story goes that the Chevalier de Mere suffered severe financial loses after incorrectly
assessing his chances of winning the following games. In order to understand his mistake he
approached Pascal to help him find his error.
First game: Roll a single die 4 times and bet on getting a 6. If you get at least one 6, you win.
Second game: Roll two dice 24 times and bet on getting double sixes. If you get at least one
pair of double sixes, you win.
Determine in each scenario if this is a game that you should play (i.e. that you have a better
chance of winning than losing). Try to calculate your probability of winning each game.
The Chevalier’s incorrect assessments of each of the games
First game: Roll a single die 4 times and bet on getting a 6. If you get at least one 6, you win.
He assessed his chances of winning as follows. The chance of getting a 6 in a single throw is 1
out of 6. Therefore, the chance of getting a 6 in four throws is 4 times 1 out of 6, i.e. 4 out of 6,
or 2 out of 3.
Second game: Roll two dice 24 times and bet on getting double sixes. If you get at least one
pair of double sixes, you win.
He assessed his chances of winning as follows. The chance of getting double sixes in one roll is
1 out of 36. Therefore, the chance of getting double sixes in 24 rolls is 24 times 1 out of 36, i.e.
24 out of 36, or 2 out of 3.
Both of his assessments were incorrect.....
Correct Assessments of the Games
First game: Roll a single die 4 times and bet on getting a 6. If you get at least one 6, you win.
Solution: Rolling a die will lead to six possible outcomes: 1, 2, 3, 4, 5, and 6. Each outcome is
equally likely. If we roll a die 4 times, there is a total possible 6 × 6 × 6 × 6 = 1296 outcomes.
Out of these, 5 × 5 × 5 × 5 = 625 of these outcomes have no 6 in them. Therefore, there are 1296
- 625 = 671 outcomes that would have a 6 in them. So, the probability of winning would be
671/1296 ≈ .5177
Second game: Roll two dice 24 times and bet on getting double sixes. If you get at least one
pair of double sixes, you win.
Solution: Rolling two dice will lead to 36 possibilities. Only one of these outcomes is a double
six. If we roll the dice 24 times, there are 3624 possibilities. The possibilities of losing are 3524
of these. The probability of losing is 3524/3624 and the probability of winning is 1 - 3524/3624 ≈
.4914
Alterations of the games
First game: Roll a single die 4 times and bet on getting a 6. If you get at least one 6, you win.
Determine the probability of winning this game if you only threw the dice three times. If you
threw the dice five times. What values for the number of rolls would give you an advantage?
Which would give you a disadvantage?
Second game: Roll two dice 24 times and bet on getting double sixes. If you get at least one
pair of double sixes, you win.
Determine the probability of winning this game if you only threw the dice twenty-three times. If
you threw the dice twenty-five times. What values for the number of rolls would give you an
advantage? Which would give you a disadvantage?
Other people’s involvement in beginning probability
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The Chevalier de Mere, Pascal, and Fermat began investigating probability in 1654.
In reality, Girolamo Cardano (1501-1576) had written the book Liber de Ludo Aleae
(Handbook of Games of Chance) more than 100 years before Pacal and Fermat solved the
Chevalier’s problem, even though the book was not published until 1663. (Remember
the name Cardano? He had explored cubic equations and imaginary numbers.)
In his book, Cardano had also discussed the Law of Large Numbers, which basically says
that the more a game is played, the more likely the experimental probability will be close
to the theoretical probability.
In 1657, the Dutch scientist Christiaan Huygens wrote “On Reasoning in a Dice Game”
in which he pulled together ideas from Pascal and Fermat and extended them to games
involving three or more players.
In his book, Huygens talked about expected value.
Consider the following game:
You pay $1 to play. The dealer has a randomly shuffled deck of cards. If you guess the next
card, you win $100. If you are wrong, you lose your dollar. The card is then re-placed into the
deck and the deck is re-shuffled. You may play the game as many times as you like. Should you
agree to play the game? Why or why not?
In this game, even though your odds are bad (1/52) when you win, you win enough to make it
worth playing. If you played 1000 times, you would win about 20 times (on average). This
means you would win about $2000 and pay about $1000 to play. Therefore, you should be up
about $1000 after playing 1000 times.
This is an example of a problem using expected value.
The influence of probability on statistics
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Probability theory applies to much more than just gambling.
Among other things, it is fundamental to the way insurance companies assess their risk
when they underwrite policies.
Jakob Bernoulli wrote about the wide range of applications of probability in his book Ars
Conjectandi (The Art of Conjecture) that was published in 1713, eight years after he died.
Bernoulli examined theoretical probability in relation to practical situations. He noted
that the assumption of equally likely outcomes was a limitation when discussing human
life spans and health. In these cases, statistical data would be more accurate.
This began the study of inferential statistics: collecting and analyzing data for
predictions.
Interesting tidbit: We now know that, because of the Law of Large Numbers, the more
insurance plans a company sells, the more likely the death rates will be as expected. In
the 18th century, however, many companies seemed to believe that the more policies they
sell, the more risk they are taking on. So they felt that selling too many policies was
dangerous for the company!