Probability theory studies the possible outcomes of given events together with their relative likelihoods and distributions. Probability deals with predicting the likelihood of future events and Probability theory enables us to find the consequences of a given ideal world.

In correspondence with Fermat Pascal laid the foundation for Probability Theory. This correspondence consisted of five letters in 1654. They considered the Dice Problem and the Problem of Points. The Dice Problem asks how many times one must throw a pair of dice before one expects a double six while the Problem of Points asks how to divide the stakes if a game of dice is incomplete.

Blaise Pascal and Pierre de Fermat began the practice of analyzing the random in response to the request of prominent gamblers. These early theorists examined games based on cards, dice and roulette wheels.

The coin toss was the simplest. There are only two probabilities -- heads and tails -- and with an evenly weighted coin they are equally likely outcomes and thus are assigned the same probability, 1 in 2.

The same analysis applies to dice, where there is a 1-in-6 probability on each roll that a five, say, will appear. With cards, there is a 1-in-52 chance that the king of hearts will be drawn from a full deck.

They came up with the Law of Large Numbers, which says simply that the proportion of performances in which some specified outcome occurs is roughly equal to the underlying probability of that outcome. If you toss a coin 1,000 times and it comes up heads only 489 times, it simply means you haven't tossed it enough and that eventually you will get a 50-50 split.This was the idea of equally probable outcomes. They realized that the 'probability' of something happening could be computed by enumerating the number of equally likely ways it could occur, and dividing this by the total number of possible outcomes of the given situation. This is what Fermat did when he figured out the 16 different (equally likely) results of tossing a coin 4 times, and then counted the ones which would result in a win for him.

Problem of Points
The original problem posed to Fermat was: Two players, A and B, each stake 32 pistoles on a three-point game. When A has 2 points and B has 1 point, the game is interrupted and cannot continue. How should the stakes of 64 pistoles be fairly distributed? Fermat's answer to this is A's 11/16 as to B's 5/16.

Pascal divided the solution into two parts. Whatever the outcome of the game, A should have at least one-half of the total (32 pistoles). Therefore, the uncertain expectation concerned only the other half, and A had a 50 percent chance of winning that. Therefore, the fair distribution would be that A received 48 pistoles (the certain 32 and one half the uncertain 32), and B received 16 pistoles.

In keeping with the legal traditions of his time, Pascal's emphasis here is on expectation and equality between the two players, the central point of his analysis, rather than on the calculation of the probability outcomes and their associated values . Hence, he eliminated probability from as much of the problem as possible, using certain gain and equity in its place.

Pascal generalized this problem to: A needs m points to win, whereas B needs n points to win; and his solution is that A's chance of winning is

sum of first m terms in the (m+n)th row of Pascal's Triangle sum of entire (m+n)th row

and B's chance of winning is the complement of this.

Independent or related events?

One of the important steps you need to make when considering the probability of two or more events occurring. Is to decide whether they are independent or related events.

A.Independent or Mutually exclusive events

The probability of throwing a double three with two dice is the result of throwing three with the first die and three with the second die. The total possibilities are, one from six outcomes for the first event and one from six outcomes for the second, Therefore (1/6) * (1/6) = 1/36th or 2.77%.

The two events are independent, since whatever happens to the first die cannot affect the throw of the second, the probabilities are therefore multiplied, and remain 1/36th.

B.Related or Mutually inclusive events
     1. What happens if we want to throw 1 and 6 in any order? This now means that we do not mind if the first die is either 1 or 6, as we are still in with a chance. But with the first die, if 1 falls uppermost, clearly It rules out the possibility of 6 being uppermost, so the two Outcomes, 1 and 6, are mutually inclusive, One result directly affects the other. In this case, the probability of throwing 1 or 6 with the first die is the sum of the two probabilities, 1/6 + 1/6 = 1/3.
     2. The probability of the second die being favourable is still 1/6 as the second die can only be one specific number, a 6 if the first die is 1, and vice versa.
     3. Therefore the probability of throwing 1 and 6 in any order with two dice is 1/3 x 1/6 = 1/18.

Converse probabilities

Often when you work out the probability of an event, you sometimes do not need to work out the probability of an event occurring you need the opposite. The probability that the event will not occur. For example, The probability of throwing a 1 on a die is 1/6 therefore the probability of a 'non-1' is (1-1/6) which equals 5/6.

The law of large numbers / "The law of averages"

The theory of probability becomes of enhanced value to gamblers when it is used with the law of large numbers. The law of large numbers states that:

“If the probability of a given outcome to an event is P and the event is repeated N times, then the larger N becomes, so the likelihood increases that the closer, in proportion, will be the occurrence of the given outcome to N*P.”

For example:

If the probability of throwing a double-6 with two dice is 1/36, then the more times we throw the dice, the closer, in proportion, will be the number of double-6s thrown to of the total number of throws. This is, of course, what in everyday language is known as the law of averages. The overlooking of the vital words 'in proportion' in the above definition leads to much misunderstanding among gamblers. The 'gambler's fallacy' lies in the idea that “In the long run” chances will even out. Thus if a coin has been spun 100 times, and has landed 60 times head uppermost and 40 times tails, many gamblers will state that tails are now due for a run to get even. There are fancy names for this belief. The theory is called the maturity of chances, and the expected run of tails is known as a 'corrective', which will bring the total of tails eventually equal to the total of heads. The belief is that the 'law' of averages really is a law which states that in the longest of long runs the totals of both heads and tails will eventually become equal.

In fact, the opposite is really the case. As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger.

An understanding of the law of the large numbers leads to a realisation that what appear to be fantastic improbabilities are not remarkable at all but, merely to be expected.

Go to Pascal's Triangle & Probability

Reference:

**1. Rogers, Ben, Pascal (London: Phoenix, 1998)P. 36-38**

2.http://www.peterwebb.co.uk/probability.htm

3.Adamson, Donald,Blaise Pascal: Mathematican,Physicist and Thinker about God ( New York,St. Martin's Press,1995)

4.http://members.fortunecity.com/kokhuitan/pascal.html

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