Linda J. Young University of Florida What is Probability? Linda J. Young University of Florida

Tali (Knucklebones) Tali is the Latin name for Knucklebones, which were called Astragaloi or Astragals by the Greeks. The Ancient Greeks originally made the pieces from the knucklebones of sheep or goats, like the ones shown below. The Romans would also make them from brass, silver, gold, ivory, marble, wod, bone, bronze, glass, terracotta, and precious gems. When tossed, the tali would each fall on one of four sides and the most common form of the game resembled modern dice.

Above is shown the backside of a bronze mirror inscribed with an image of Venus playing Tali with Pan. This mirror dates from 350 BC and comes from Greece, where Venus was known as Aphrodite. In 350 BC players in both Greece and Rom likely still played with astragali, but more and more they changed to using dice.

The Start of Modern Probability Antoine Gambaud, nicknamed “the Chevalier de Méré, gambled frequently to increase his wealth. He bet on a roll of a die that at least one 6 would appear during a total of four rolls. From past experience, he knew that he was more successful than not with this game of chance. Tired of his approach, he decided to change the game. He bet that he would get a total of 12, or a double 6, on 24 rolls of two dice. Soon he realized that his old approach to the game resulted in more money.

In 1654, Chevalier de Méré asked his friend Blaise Pascal why his new approach was not as profitable.                                        The problem proposed by Chevalier de Méré was the start of famous correspondence between Pascal and Pierre de Fermat. They continued to exchange their thoughts on mathematical principles and problems through a series of letter, leading them to be credited with the founding of probability theory.

Although several mathematicians developed probability from the time of Pascal and Fermat, Andrey Kolmogorov developed the first rigorous approach to probability in 1933. He built up probability theory from fundamental axioms in a way comparable with Euclid’s treatment of geometry.

If I make everything predictable, human beings will have no motive to do anything at all, because they will recognize that the future is totally determined. If I make everything unpredictable, will have no motive to do anything as there is no rational basis for any decision. I must therefore create a mixture of the two. (from E.F. Schumacher, Small is Beautiful ) LORD

Deterministic Models Models of reality Examples: Area of a rectangular plot of land: A = L W where L is the length of the plot and W is its width Force with which a football player hits his opponent: F = mA where m is the mass and A is acceleration

Probabilistic Models Formal and informal models Deals with uncertainty Plays an important role in decision making in day-to-day activities No statistics or stochastic processes without probability

Probability Probabilistic experiments go beyond coin tossing, picking cards, throwing dice, etc. Probability helps us to understand better the events surrounding us. Look around and see most things in life have uncertainty. We accept some uncertainty with no real concern. Weather, time to reach school (work), prices of goods, regular fluctation in stocks, etc. Breakdown of cars, outage of electricity or gas, crash of stock markets, etc.

Probability: Foundation for Decision Making How do the insurance companies determine the premiums? How do the manufacturing companies determine the warranty period? How do the manufacturers decide on the number of units to make? How do the supermarkets decide on the number of counters to open?

Probability: Foundation for Decision Making (continued) How do the package delivery companies offer the guarantee and charge? How do the package delivery companies schedule their drivers, fleet, etc? How do the airlines schedule their crew, fleet, etc? How is the jury selected? How do the casinos determine the pay out for the odds in a bet?

Probability: Foundation for Decision Making (continued) Why is it that, if you go to a bank or post office, you see there is only one queue in front of many tellers? Why is it that, in super markets, you see several (parallel) queues? Have you ever wondered, when you call your friend over the phone, how in spite of not having a “direct” connection, you get connected without delay? DNA matching (especially in crime related activities) is important in a judicial process. Have you wondered how probability plays a role here?

What is Probability? One approach: long-run or limiting relative frequency Suppose that an experiment is conducted n times. Let n(A) denote the number of times the event A occurs Intuitively it suggests that P(S) can be approximated with n(A)/n N(A)/n will approach P(A) as n approaches infinity