The Story of Blaise Pascal

In 1654, a French nobleman, Chevalier de Méré, challenged Pascal to solve a puzzle known at the time as the “problem of points”.   The problem, first posed by an Italian monk in the late 1400s, had remained unsolved for nearly two hundred years.  The issue in question was to decide how the stakes of a game of chance should be divided if that game were not completed for some reason.   The example used in the original publication[1] referred to a game of “balla” where six goals were required to win the game.  If the game ended normally, the winner would take all.  But what if the game stopped when one player was in the lead by five goals to three?  

In seeking a solution to the problem, Pascal entered into correspondence with the lawyer and mathematician Pierre de Fermat.  Between the two of them they laid the foundations of modern probability theory.  What Fermat and Pascal realized was that the solution came from listing all of the possibilities and then counting the proportion of the time that each player would win.  In order to understand their solution, we can simplify the problem to a coin-tossing game with two players.  The game ends when the coin has come up heads or tails twice.  We will say that Player 1 wins in the first instance, while Player 2 wins in the second.  If the game reaches its natural conclusion, either Player 1 or Player 2 will take the entire amount bet.  In the spirit of the problem of points, we can ask what happens if the game is interrupted after one toss that has come up heads?

In his letters to Fermat, Pascal shows how to figure out the answer.  First, you need to list all of the possibilities from the very beginning.  That means asking what the maximum number of times is that the coin would need to be tossed to insure a winner.  Here we know that the answer is three.  We then look at all the combinations of heads and tails that are possible in three tosses – this is a total of eight.  In this case, though, we know that the first toss was heads, and so we only need to consider the four cases that begin with a heads, and can ignore the remaining four. Using H to signify a heads and T a tails, we can write these four possibilities as (HHH), (HHT), (HTH), and (HTT). The first three of these four contain at least two heads and so Player 1 is the winner, while the last one contains one head and two tails, and Player 2 wins.   If the coin has an equal chance of coming up head or tails, then each of these four possibilities is equally likely.  This means that, if the coin comes up heads on the first flip, then Player 1 has a 3 in 4 chance of winning.  If this game is interrupted, the two players should share the pot ¾ for Player 1 and ¼ for Player 2.  But the only way we could figure this out was to count all of the possibilities.

From this approach, Pascal derived more general results and developed rules of probability. (He then abandoned mathematics for a quieter life in a Parisian monastery.)  While Pascal’s contribution to probability theory was undoubtedly substantial, it was just the beginning.   Importantly, his analysis didn’t stretch to more realistic situations where a finite number of equally likely possible outcomes could not be listed.  The weather is such an example.[2]

In the early 18th century, Jacob Bernoulli, who stressed the role of statistical sampling in dealing with uncertainty, addressed problems with a potentially infinite number of outcomes.  Through his law of large numbers, Bernoulli sought to provide a formal proof of the idea that uncertainty decreased as the number of observations increased.   Another key development of that century was the discovery by Abraham De Moivre of the normal curve – the fact that random drawings would distribute themselves in a bell shape around their average value.   Although theories relating to risk and uncertainty have continued to develop, the contributions of Pascal, Bernoulli and De Moivre remain pivotal to our understanding of risk.

Pascal and the Problem of Points

Blaise Pascal’s invention of probability can be traced to 1654, when a French nobleman named Chevalier de Méré, challenged Pascal to solve a puzzle known at the time as the “problem of points”.   The problem, first posed by an Italian monk in the late 1400s, had remained unsolved for nearly two hundred years.  The issue in question was to decide how the stakes of a game of chance should be divided if that game were not completed for some reason.   The example used in the original publication[3] referred to a game of “balla” where six goals were required to win the game.  If the game ended normally, the winner would take all.  But what if the game stopped when one player was in the lead by five goals to three?  

In order to understand their solution, we can simplify the problem to coin-tossing game with two players.  The game ends when the coin has come up heads or tails twice.  We will say that Player 1 wins in the first instance, while Player 2 wins in the second.  If the game reaches its natural conclusion, either Player 1 or Player 2 will take the entire amount bet.  In the spirit of the problem of points, we can ask what happens if the game is interrupted after one toss that has come up heads?

In his letters to Fermat, Pascal shows how to figure out the answer.  We will use the technique from Table A.  Finding the solution starts by listing all of the possibilities. That means asking what the maximum number of times is that the coin would need to be tossed to insure a winner.  Here we know that the answer is three.  We then look at all the combinations of heads and tails in three tosses – this is a total of eight. 

All possible sequences of heads and tails from three flips of a coin are listed in Table 5.2.  If the coin has an equal chance of coming up head or tails, these are all equally likely, and so have a probability of 1/8.

We can now use this to solve the problem.  Remember we want to know the chances of Player 1, who wins when two heads come up before two tails, winning if the first flip was Head. This immediately eliminates possibilities 5 through 8, where the first flip is a Tail, and focuses our attention on possibilities 1 through 4.  Notice two things about these first four rows.  First, all four cases have of an equal probability of occurring, and second, in 3 of the 4 Player 1 wins.  This gives us the answer to the question.  If this game is interrupted, the two players should share the pot ¾ for Player 1 and ¼ for Player 2.  But the only way we could figure this out was to count all of the possibilities.