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 rotate
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