Matthew Rossi

Probability theory is the study of the likelihood of an occurrence of random events in order to predict future behaviors of a system (2). The principles of probability are widely used. In genetics, for example, probability is used to estimate the likelihood of gene distribution from one generation to the next. In business, insurance companies use the principles of probability to determine risk groups. Probability is closely related to statistics since uncertainty always exists when statistical predictions are being made. A number between 0 and 1 represents the probability of an outcome (1). The probability of an impossible event is 0. Where as the probability of something that is certain to occur is 1. The theory of probability is recognized as being developed by Blaise Pascal with help from his friend Pierre de Fermat.

Blaise Pascal was born at Clermont, France on June 19, in 1623. He was the third child of Etienne Pascal, and his only son. Blaise was only 3 when his mother died (3). In 1631, his family moved to Paris to carry on the education of Blaise, who had already displayed exceptional ability. Pascal was home taught, and to ensure that he was not overworked, his father decided that his studies would only involve the languages, and should not include any mathematics. At the age of twelve, Pascal demonstrated to his tutor an interest in geometry. He was stimulated by the subject, and gave up his playtime and chose to study geometry instead. In a few weeks, he discovered the many properties of geometric figures, in particular, that the sum of the angels of a triangle equals180 degrees. Impressed by Pascal’s display, his father gave him a copy of Euclid's Elements, which Pascal read and soon mastered (5).

At the age of fourteen, Pascal was admitted to the weekly meetings of French geometricians, which ultimately became the French Academy. His first work, Essay on Conic Sections, was published in February of 1640. Between 1642 and 1645, he invented the first digital calculator that helped his father with his work of collecting taxes. However, there were problems with the machine since it did not work well with the design of French currency, so it never became successful (3). Pascal then began to turn his attention toward analytical geometry and physics. He repeated many of the experiments of Toricelli, the inventor of the barometer, and confirmed the theory of barometrical variations. Then in the middle of this research, in 1650, Pascal abandoned his mathematical pursuits to study religion.

After his father's death in 1653, Pascal had to administer his father's estate, and this reignited his interest in mathematics. It was at this point that he invented the arithmetical triangle (now known as Pascal's triangle), and together with Fermat created the theory of probability. On November 23, 1654, he had a near fatal accident with a horse carriage, but was saved when the traces broke. This incident caused him to again turn towards religion and pledge his life to Christianity. He then retired to Port Royal, where he continued to live until his death in 1662. From 1657 to 1658, he wrote his most famous work, Pensées, which is a collection of his personal thoughts on human suffering and faith in God (3). He rationalized believing in God with this argument, "if God does not exist, one will lose nothing by believing in Him, while if He does exist, one will lose everything by not believing." Pascal died on August 19, 1662, at the young age of 39 because he had injured his health as a teenager by his incessant studying (5).

Pascal developed the theory of probability after his friend, Antoine Gombaud; a French nobleman with an interest in gambling, confronted Pascal with a question regarding a popular dice game. The question was this. Two players want to leave a game before finishing. With their scores given, it is desired to find what proportion they should divide the winnings. During 1654 Pascal and Fermat wrote several letters back and forth to each other until they agreed upon a general theory of probability (5). Here is Pascal's explanation.

After that point, many mathematicians contributed to the theory of probability. There was much appeal for this branch of mathematics because of its ties with games of chance. In 1657, a Dutch scientist, Christian Huygens published the first book on probability, which was filled with gambling related problems. The subject developed quickly during the eighteenth century with contributions from Jakob Bernoulli and Abraham de Moivre. Then in 1812, Pierre de Laplace introduced many new ideas and mathematical techniques demonstrating many scientific and practical problems for probability theory. Some of the applications for probability theory developed in the nineteenth century included theory of errors, actuarial mathematics, and statistical mechanics. In 1933, a Russian mathematician, A. Kolmogorov developed a comprehensive definition of probability theory, which became the basis for the modern theory. Probability is now widely used in the fields of genetics, psychology, economics, and engineering (4).

The theory of probability can be shown through the example of a simple coin game. Take three quarters, color both sides of one quarter, and one side of another quarter black. 3 of the 6 sides of the quarters are black, and three are silver. Now, shuffle the quarters and pull one out so that only one side is seen. If the side showing is black, bet the person you are playing that the other side is also black. If the side showing is silver, guess that the other side is silver. There is a two-thirds probability that you are correct. To begin with, two of the coins have the same color on both sides and only one has one side black, one side silver. So there is a two out of three chance of picking a quarter with the same color on both sides, and only a one out of three chance of picking a quarter with two different colored sides. By guessing the same color on the other side of the quarter, one is really betting on the two-thirds probability of picking a quarter with the same color on both sides.

During the late 1850’s, Gregor Mendel used the general rules of probability to explain the basic principles of heredity by breeding green peas in planned experiments. A heritable feature is called a character, and variants of a character are called traits. Traits occur because of different versions of genes on the chromosomes of every cell of a living organism. Different versions of genes are called alleles. For most genes, two alleles exist, one dominant over the other. Each organism has two alleles. If an organism has two different alleles for the same gene, the dominant allele will be expressed in the phenotype of the organism, while the recessive allele will not. The phenotype of an organism is its physical appearance, as in Mendel’s peas, they could either have a round shape (S), or a wrinkled shaped (s) (1).

Alleles are passed on from parents to offspring after they are bred. Each parent will pass on one of its two alleles. Which allele is passed on depends on probability. For seed shape, a parent could either have two dominant alleles (SS) for round shape, two recessive alleles (ss) for wrinkled shape, this condition is called homozygous. If a pea plant has one of each allele (Ss) in which the round shape will be expressed, this is called heterozygous. Homozygous pea plants will only pass on their one allele, so the probability of them passing on their one allele is 1. Heterozygous pea plants on the other hand will have a ½ probability of passing on the dominant allele (S), and a ½ probability of passing on the recessive allele (s).

Mendel was able to show how these alleles are passed on by performing a breeding experiment with true-breeding pea plants. True-breeding means the plants are homozygous for a particular trait, in this case, seed shape. He began by crossing a pea plant that was homozygous for round seeds (SS) with a pea plant that was homozygous for wrinkled seeds (ss). All of the offspring in the first generation (F1) will have round seeds, but be heterozygous (Ss). All of the offspring are heterozygous because each parent passes on its one allele, so the offspring gets a dominant allele from one parent and a recessive allele from the other. The pattern of inheritance is then shown when two of the heterozygous plants are crossed (1).

Two heterozygous (Ss) plants are then crossed and the result is a 3 to 1 ratio of round seeds to wrinkled seeds. This phenomenon can be explained by using the rule of multiplication and the rule of addition to predict which characteristics the second offspring generation (F2) will exhibit. The rule of multiplication involves computing the probability for each independent event, then multiplying these individual probabilities to obtain the overall probability of these events occurring together. The rule of addition is the sum of separate probabilities if an event can occur in more than one way. The probability that the first pea plant will pass on the dominant allele is ½. The probability that the second pea plant passed on the dominant allele is also ½. So ½ x ½ = ¼. ¼ of the second generation of plants will be homozygous dominant for round seeds (SS). ¼ will also be homozygous recessive (ss), because the probability is the same for the recessive allele, ½ x ½. As previously stated, the probability of the first plant passing on the dominant allele is ½, and the probability of the second plant passing on the recessive allele is ½. Conversely, the first plant could give the recessive allele and the second plant could give the dominant allele, which will also product a heterozygous plant. Therefore, the proportion of heterozygous plants (Ss) for round seed shape will be (½ x ½) + (½ x ½) = ½. The ¼ for homozygous dominant (SS) plus the ½ for heterozygous (Ss) shows that ¾ will have the dominant allele (S), and have round seed shape. The remaining ¼ will have wrinkled seeds because they will be homozygous recessive (ss). This relationship can be seen more easily through the use of a Punnett Square (1).

 Mendel performed this experiment with hundreds of pea plants while tracking the distribution of 6 other traits, such as flower color. With each trait the ratio was 3 to 1, dominant to recessive, as expected by the rules of probability. These same principles of probability are used today in genetics to solve more complicated problems involving many genes with more difficult modes of expression.

Probability is a very interesting branch of mathematics because it is used to predict the likelihood of random events. It is a deductive science that studies uncertain quantities related to random events (2). Many people seem to enjoy the ideas of probability because it is related to so many games that usually involve money. By playing the odds right, someone could win big, either at a card table or on Wall Street. Probability makes random events look like very predictable ones.

1. Campbell, Neil, Jane Reece, Lawerence Mitchell. Biology fifth edition.

2.Interactive Mathematics Miscellany and Puzzles, Probability.

3. Pascal. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html. (12/5/99)

4. A Short History of Probability. From Calculus, Volume II by Tom M. Apostol

5.Wilkins, D. R. Blaise Pascal (1623 – 1662)

Peer review by: Bret Willet

Punnett Square from http://web.psych.ualberta.ca/~msnyder/Academic/p104/lec1/17.html (12/5/99)

Clipart from http://www.caboodles.com/clipart/themes/index.html (12/5/99)