Probability Essay, Research Paper

Probability

Probability is the branch of mathematics that deals with measuring or determining the

likelihood that an event or experiment will have a particular outcome. Probability is based on

the study of permutations and combinations and is also necessary for statistics.17th-century

French mathematicians Blaise Pascal and Pierre de Fermat is usually given credit to the

development of probability, but mathematicians as early as Gerolamo Cardano had made

important contributions to its development. Mathematical probability began when people tried to

answer certain questions that was in games of chance, such as how many times a pair of dice

must be thrown before the chance that a six will appear is 50-50. Or, in another example, if two

players of equal ability, in a match to be won by the first to win ten games, is the other player

suspend from play when one player has won five games, and the other seven, how should the

stakes be divided?

Permutations and combinations are the arrangement of objects. The difference between

permutations and combinations is that combinations pays no attention to the order of

arrangement and permutations includes the order of arrangements of objects.

Permutations is the idea of permuting n number of objects. For example, when n = 3

and the objects are x, y, and z, the permutations or the number of arrangements are xyz, xzy,

yzx, yxz, zyx, and zxy. That means that there is 6 ways that x, y, and z can be arrange. Another

way of finding out the answer is using factorial. Here there are 6 permutations, or 3 ? 2 ? 1 = 3!

The answer 3! is read as three factorial and that tells you all the positive integers numbers

between 1 and 3. The formula for the factorial is:

n ! = n ? (n – 1) ? ? ? 1 permutations

For example, if there are n teams in a league, and ties are not possible, then there are

n ! possible team rankings at the end of the season. A slightly more complicated problem

would be finding the number of possible rankings of the top r number of teams at the end of a

season in a league of n teams. Here the formula is

nPr = n ? (n – 1) ? ? ? (n – r + 1) = n !/(n – r) !

so that the number of possible outcomes for the first four teams of an eight-team league is

8P4 = 8 ? 7 ? 6 ? 5 = 840.

Now what if we weren?t interested in the order in which the top four teams finished,

but interested in only about the number of the possible combinations of teams that could be in

the top four positions in the league at the end of the season. This is what finding a four-object

combinations out of an eight-object set or 8C4. In general, an r-combination of n objects(n is

greater than r) is the number of distinct groupings of r elements pulled from a set of n

elements. The formula for this number, written (nCr) or (nPr)/r !. For example, the

2-combinations of the three elements a, b, and c are ab, ac, and bc or can be written as 3C2 =

3. The general formula for (nCr) is:

n !/[r !(n - r) !] (This expression can also me written as (nr))

If repetitions or a given element can be chosen more than once is permitted, then the

last example would also include aa, bb, and cc which adds up to 6. The general formula for the

number of r-combinations from an n-element set is

(n + r – 1) !/[r !(n - 1) !]

For example, if a teacher must make a list containing three names from a class of 15,

and if the list can contain a name two or three times and order does not matter, then there are

(15 + 3 – 1) !/[3 !(15 - 1) !] = 680 possible lists. In the case of r-permutations with repetition

from an n-element set, the formula is nr. For example, to the six 2-permutations of a, b, c

without repetitions (ab, ba, ac, ca, bc, and cb) are added the three with repetitions (aa, bb, and

cc), for a total of 9, which is equal to 32. Thus, if two prizes are to be awarded among three

people, and it is possible that one person could receive both prizes, then nine possible

outcomes exist.

Finally, suppose there are n1 objects of one type, n2 of another type, on to n3 objects of

some third type. Let n = n1 + n2 + ? + n2. In how many ways can these objects be arranged

and also keeping order? The answer is n !/(n1 !n2 ! ? n3 !), One example is how many letters

of the word banana can be arranged? 60 letters because 6 !/(3 !2 !1 !) = 60. This is also the

coefficient of x3y2z1 in (x + y + z)6.

The most common use of probability is used in statistical analysis. For example, the

probability of throwing a 7 in one throw of two dice is 1/6, and this answer means that if two

dice are randomly thrown a very large number of times, about one-sixth of the throws will be 7s.

This method is most commonly used to statistically determine the probability of an outcome that

cannot be tested or is impossible to obtain. So, if long-range statistics show that out of every 100

people between 20 and 30 years of age, 42 will be alive at age 70, the assumption is that a person

between those ages has a 42 percent probability of surviving to the age of 70.

Today, almost every use probability in their everyday especially people who gamble.

In Nevada casinos you will find the table game of Chuck-a-Luck in which there are three dice in

a rotating cage having an hourglass shape. You bet on one or more of the six possible numbers

and the cage is rotated. If your number comes upon one die you win back your money. If it

comes up on two dice, you win twice the amount of your bet and if it comes up on all three dice

you win three times the amount you bet.

People who barely remember having studied probability in high school algebra courses

sometimes reason as follows: “The probability of my number coming up on one die is 1/6, so the

probability of its coming up on one of the three is three times this, or 1/2. That is a fair bet right

there so obviously this game favors the player.” This is a con game in which the player is

conning himself. What the player has forgotten is that his simple rule for addition of

probabilities applies only if the possible outcomes are mutually exclusive. Such is not the case

here, for your number coming up on one of the dice does not prevent it from coming up on

others as well. When the events are not mutually exclusive one must subtract the probabilities of

repeats. Rather than doing this, let’s run through the 63 = 216 possible ways the dice can come

up and court house cases in which a particular number comes up exactly once.

Let’s say the number comes up on die A. There are 5 ways it can not also come up on die

B and 5 ways it can not also come up on die C, so the number of cases in which the number

comes up only on die A is5 x 5 = 25. The number of ways it can come up exclusively on die B is

the same, and also is the same for die C, so the number of ways a particular number can come up

once only on any of the three dice is 3 x 25 = 75. The probability that this will happen is 75/216

which is approximately equal to 0.35 and is considerably less than one-half. The number of ways

a particular number can come up on exactly two dice is equal to the number of ways it can fail to

come up on the third. This is five times the number of dice it can fail to come up on, or15. The

probability of your number coming up on exactly two dice is15/216. There is only one way in

which your number can come up on all three dice, so the probability of this happening is 1/216.

Now lets calculate the values of the winning outcomes. If your number comes up exactly once

you get two units back, so the value of this outcome is 2 x 75/216 = 150/216.If it comes up twice

you get three units back, so the value of this outcome is 3 x 15/216 = 45/216. If it comes up on

all three you get four units back, so this outcome has a value of 4/216. Adding these up, we get

the value of the winning outcomes as 199/216 units. All the other possible outcomes are losers

with the value of -1 unit. Adding the two, we see that the net value of the game to the player is -

17/216 or approximately -0.0787. This means that the house has an edge of nearly 8%, which is

more than that of some of the other games. That is why some people will never figure out why

they don?t consistently win at Chuck-a-Luck.

Here is a proposition that someone actually tried to sell my father on. He was a salesman

for a filing service for entries in a U. S. Government lottery for oil exploration leases on federal

land. His propositions was “Maybe there is only one chance in 100 of your entry being selected

in the lottery, but if you file 100 entries, then you are SURE to win. ?When he challenged him on

this he didn’t try to convince my father, so he knew that it was wrong. Later on he told me that

the probability of your winning the lottery is equal to the number of entries you file divided by

the total number of entries filed. You never are sure to win unless nobody else files. The

salesman tried to use common sense by turning it into an abstract problem in probabilities. Even

treating the situation in an abstract way, he was wrong. Let be the probability of your winning at

least once in n tries. If p is the probability of winning in a single try, then assuming that the

outcomes are independent of each other

w = 1 – (1 – p)n

If you want to know the number of tries needed to attain a certain probability of winning you can

use the formula:

n = ln (1 – w) / ln (1 – p)

with n rounded up to the next larger integer. This formula uses natural logarithms(the power to

which a number, called the base, must be raised in order to obtain a given positive number. For

example, the logarithm of 100 to the base 10 is 2, because 10 2 = 100. Common logarithms use

10 as the base; natural, or Napierian, logarithms use the number e as the base) but you can use

any kind of logarithm. Since both p and w must lie in the range 0 to 1, both the numerator and

denominator will be negative. If w is set very close to 1, then n becomes very large. Industrialists

Roy Kroc and Ross Perot know this formula. It means that if you want to be pretty sure of

winning you have to be persistent.

Mathematical probability is widely used in the physical, biological, and social sciences

and in industry and commerce. It is applied in many areas like genetics, quantum mechanics,

and insurance. It also involves deep and important theoretical problems in pure mathematics

and has strong connections with the theory, known as mathematical analysis, that developed

out of calculus.

Bibliography

1) Introduction To Statistics by Susan F. Wagner (Book)

2) Student Reference Library by Mindscape (CD-Rom)

3) http:\\www.encyclopedia.com (Internet) Search Phrase: Probabilty, Permutations,

Combinations

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