**PROBABILITY**

**brief history**

**What is Probability?**

It is the way of expressing knowledge of belief that an event will occur on a

chance.

It is originated from the Latin word meaning approval.

It is mainly the mathematics of

gambling

because it is the collection of probability applications encountered in games of chance.

"A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians,

Blaise Pascal and Pierre de Fermat

. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite.

This problem and others posed by de Méré led to an exchange of letters between Pascal and Fermat in which the fundamental principles of probability theory were formulated for the first time. Although a few special problems on games of chance had been solved by some Italian mathematicians in the 15th and 16th centuries, no general theory was developed before this famous correspondence.

The Dutch scientist Christian Huygens, a teacher of Leibniz, learned of this correspondence and shortly thereafter (in 1657) published the first book on probability; entitled De Ratiociniis in Ludo Aleae, it was a treatise on problems associated with gambling. Because of the inherent appeal of games of chance, probability theory soon became popular, and the subject developed rapidly during the 18th century. The major contributors during this period were Jakob Bernoulli (1654-1705) and Abraham de Moivre (1667-1754).

In 1812 Pierre de Laplace (1749-1827) introduced a host of new ideas and mathematical techniques in his book, Théorie Analytique des Probabilités. Before Laplace, probability theory was solely concerned with developing a mathematical analysis of games of chance. Laplace applied probabilistic ideas to many scientific and practical problems. The theory of errors, actuarial mathematics, and statistical mechanics are examples of some of the important applications of probability theory developed in the l9th century.

Like so many other branches of mathematics, the development of probability theory has been stimulated by the variety of its applications. Conversely, each advance in the theory has enlarged the scope of its influence. Mathematical statistics is one important branch of applied probability; other applications occur in such widely different fields as genetics, psychology, economics, and engineering. Many workers have contributed to the theory since Laplace's time; among the most important are Chebyshev, Markov, von Mises, and Kolmogorov.

GIROLAMO CARDANO

"The father of Probability"

DEFINITIONS

EXPERIMENT- is a situation involving chance or probability that leads to results called outcomes

OUTCOME- is the result of a single trial of an experiment.

EVENT- is one or more outcomes of an experiment.

Girolamo Cardano is sometimes known by his Latin name, Cardan. He was an illegitimate child of a lawyer in Milan, whose expertise in mathematics was such that he was consulted by Leonardo da Vinci on questions of geometry. Cardano at first became his father's assistant, but began to think about an academic career after learning mathematics from his father. He studied medicine, and was a brilliant student. But he was outspoken and highly critical, so he was not well liked.

Cardano squandered the small bequest from his father and turned to gambling to make a living. Cardano's understanding of probability meant he had an advantage over his opponents and, in general, he won more than he lost. Gambling became an addiction that was to last many years and rob Cardano of valuable time, money and reputation.

Cardano was awarded his doctorate in medicine in 1525. He set up a small, and not very successful, medical practice in Sacco, where he married. He repeatedly applied to the College of Physicians in Milan but was not allowed membership due to his reputation and his ignoble birth. Unable to practise medicine, Cardano reverted again to gambling to pay his way. Things went so badly that he was forced to pawn his wife's jewellery and even some of his furniture.

Cardano was fortunate to obtain the post of lecturer in mathematics in Milan which gave him plenty of free time, and he used some of this to treat a few patients, despite not being a member of the College of Physicians. Cardano achieved some near miraculous cures and his growing reputation as a doctor led to his being consulted by members of the College, to which he was eventually admitted in 1539.

In addition to Cardano's major contributions to algebra he also made important contributions to probability, hydrodynamics, mechanics and geology. Cardano made the first foray into the untouched realm of probability theory. Cardano also published 2 encyclopaedias of natural science, which contain a little of everything, from cosmology to the construction of machines, from the usefulness of natural sciences to the evil influence of demons, from the laws of mechanics to cryptology.

Cardano is reported to have correctly predicted the exact date of his own death but it has been claimed that he achieved this by committing suicide.

As Cardano was at the height of his fame, he received what he called his "crowning misfortune". Cardano's eldest son secretly married a girl whom he later poisoned. Following his arrest, he confessed to the crime and he was executed. This was a blow from which Cardano never recovered. As the father of a convicted murderer, Cardano became a hated man. In 1570, Cardano himself was put in jail on the charge of heresy for casting the horriscope of Jesus Christ. On his release a few months later, he was forbidden to hold a university post and barred from further publication of his work.

THE PROBABILITY OF EVENT

P(E)=

NUMBER OF EVENT OUTCOMES

-------------------------------------------------------

TOTAL NUMBER OF POSSIBLE OUTCOMES

3 TYPES OF PROBAILITY

1. THEORITICAL PROBABILITY- it is the likeliness of an event happening based on all the possible outcomes.

2. EXPERIMENTAL PROBABILITY- probability based on an experiment written as a ratio comparing the number of times the event occurred to the number of trials.

3. SUBJECTIVE PROBABILITY-Numeric measure probability that reflects the degree of a personal belief in the likelihood of an occurrence.

EXAMPLES

If we toss a fair coin, what is the probability that a tail will show up?

Solution:

a. Tossing a tail is the favorable outcome here.

b. When you toss a coin there are only 2 possible outcomes: a Head or a Tail

c. So the options for tossing a tail are 1 out of 2.

d. We can also represent probability as a decimal or as a percent.

A coin is tossed 60 times. 27 times head appeared. Find the experimental probability of getting heads.

Solution:

Step 1: Experimental probability = number of times the event occurs / total number of trials.

Step 2: Number of times heads appeared = 27.

Step 3: Total number of experiments = 60.

Step 4: So, experimental probability of getting a head = 27/60= 9/20.

Two coins are tossed, find the probability that two heads are obtained.

Note: Each coin has two possible outcomes H (heads) and T (Tails).

Solution to Question 2:

The sample space S is given by. S = {(H,T),(H,H),(T,H),(T,T)}

Let E be the event "two heads are obtained".

E = {(H,H)}

We use the formula of the classical probability.

P(E) = n(E) / n(S) = 1 / 4

postulates on probability

1.) 0 <= P(A) <= 1

2.) P(A) = P(S) = 1

3.) Given the events A & B are subset of S

A

B

a. Mutually Exclusive Events

P(A U B)= P(A)+P(B)

b. Non Exclusive Events

P(A U B)= P(A)+P(B)-P(A B)

A

B

d

S

S

4.) P(A') = 1 - P(A)

A

S

5.) For independent events A&B which can occur simultaenously P(A B) = P(A) * P(B)

U

U

examples

1. Probability of getting a 5 or a 6 in a single throw of s die.

SOLUTION:

P(5 U 6)= P(5) + P(6)

=1/6+1/6

=1/3

2. Probability of getting an even or a prime number in a single throw of die.

P(E U P) = P(E)+P(P)-P(E P)

= 1/2 + 1/2 - 1/6

= 5/6

U

independent events

What ever happens in one event has absolutely nothing to do with what will happen next because:

a. The two events are unrelated

b. You repeat an event with an item whose number will not change

c. You repeat the same activity but you replace the item that was removed.

P (A B)=P(A)-P(B)

U

DEPENDENT EVENTS

What happens during the second event depends upon what happened before.

P(A B) = P(A) * P(B)

U

conditional probability

It is the probability of the event given that another event has occurred

Let A & B be two events in a sample space S. The conditional probability of an event B occurred given that event A has occurred is defined to be:

P(A l B) =

P(A B)

---------------

P(A)

U

P(A) should not be equal to 0

SAMPLE PROBLEM:

Two dice are rolled in succession and the 1st dice shows a three. Knowing this, find the probability that the sum shown by the dice is five, six or seven.

Solution:

P(A)= 1/6

P(A B)=3/12

P(A B)

U

U

-------

P(A)

=

1/12

-------

1/6

=

1/2