Transcript 2003 ENAR Student Awards

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originally made the pieces from the knucklebones of sheep or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