Welcome to 6.041/6.431, a subject on the modeling and analysis of random phenomena and processes, including the basics of statistical inference. Nowadays, there is broad consensus that the ability to think probabilistically is a fundamental component of scientific literacy. For example: The concept of statistical significance (to be touched upon at the end of this course) is considered by the. This site is the homepage of the textbook Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. 5 synonyms of probability from the Merriam-Webster Thesaurus, plus 17 related words, definitions, and antonyms. Find another word for probability. Probability: the quality or state of being likely to occur.

Probability is the branch of mathematics that studies the possible outcomes of given events together with the outcomes' relative likelihoods and distributions. In common usage, the word 'probability' is used to mean the chance that a particular event (or set of events) will occur expressed on a linear scale from 0 (impossibility) to 1 (certainty), also expressed as a percentage between 0 and 100%. The analysis of events governed by probability is called statistics.

There are several competing interpretations of the actual 'meaning' of probabilities. Frequentists view probability simply as a measure of the frequency of outcomes (the more conventional interpretation), while Bayesians treat probability more subjectively as a statistical procedure that endeavors to estimate parameters of an underlying distribution based on the observed distribution.

A properly normalized function that assigns a probability 'density' to each possible outcome within some interval is called a probability density function (or probability distribution function), and its cumulative value (integral for a continuous distribution or sum for a discrete distribution) is called a distribution function (or cumulative distribution function).

A variate is defined as the set of all random variables that obey a given probabilistic law. It is common practice to denote a variate with a capital letter (most commonly ). The set of all values that can take is then called the range, denoted (Evans et al. 2000, p. 5). Specific elements in the range of are called quantiles and denoted , and the probability that a variate assumes the element is denoted .

Probability 233 Definition

Probabilities are defined to obey certain assumptions, called the probability axioms. Let a sample space contain the union () of all possible events , so

and let and denote subsets of . Further, let be the complement of , so that

Then the set can be written as

where denotes the intersection. Then

(4)

(6)

(8)

where is the empty set.

Let denote the conditional probability of given that has already occurred, then

(10)

(12)

(14)

The relationship

holds if and are independent events. A very important result states that

which can be generalized to

Probability 233 Math

SEE ALSO:Bayes' Theorem, Conditional Probability, Countable Additivity Probability Axiom, Distribution Function, Independent Statistics, Likelihood, Probability Axioms, Probability Density Function, Probability Inequality, Statistical Distribution, Statistics, Uniform DistributionREFERENCES:

Evans, M.; Hastings, N.; and Peacock, B. StatisticalDistributions, 3rd ed. New York: Wiley, 2000.

Everitt, B. Chance Rules: An Informal Guide to Probability, Risk, and Statistics. Copernicus, 1999.

Goldberg, S. Probability:An Introduction. New York: Dover, 1986.

Keynes, J. M. ATreatise on Probability. London: Macmillan, 1921.

Mises, R. von MathematicalTheory of Probability and Statistics. New York: Academic Press, 1964.

Mises, R. von Probability,Statistics, and Truth, 2nd rev. English ed. New York: Dover, 1981.

Mosteller, F. FiftyChallenging Problems in Probability with Solutions. New York: Dover, 1987.

Mosteller, F.; Rourke, R. E. K.; and Thomas, G. B. Probability:A First Course, 2nd ed. Reading, MA: Addison-Wesley, 1970.

Nahin, P. J. Duelling Idiots and Other Probability Puzzlers. Princeton, NJ: Princeton University Press, 2000.

Neyman, J. FirstCourse in Probability and Statistics. New York: Holt, 1950.

Rényi, A. Foundationsof Probability. San Francisco, CA: Holden-Day, 1970.

Ross, S. M. A First Course in Probability, 5th ed. Englewood Cliffs, NJ: Prentice-Hall, 1997.

Ross, S. M. Introduction to Probability and Statistics for Engineers and Scientists. New York: Wiley, 1987.

Ross, S. M. AppliedProbability Models with Optimization Applications. New York: Dover, 1992.

Ross, S. M. Introductionto Probability Models, 6th ed. New York: Academic Press, 1997.

Probability 233 Probability

Székely, G. J. Paradoxes in Probability Theory and Mathematical Statistics, rev. ed. Dordrecht, Netherlands: Reidel, 1986.

Todhunter, I. A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace. New York: Chelsea, 1949.

Weaver, W. LadyLuck: The Theory of Probability. New York: Dover, 1963.

Referenced on Wolfram Alpha: ProbabilityCITE THIS AS:

Weisstein, Eric W. 'Probability.' FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Probability.html

Introduction

Probability is the likelihood or chance of an event occurring.

Probability =	the number of ways of achieving success
the total number of possible outcomes

For example, the probability of flipping a coin and it being heads is ½, because there is 1 way of getting a head and the total number of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .

• The probability of something which is certain to happen is 1.
• The probability of something which is impossible to happen is 0.
• The probability of something not happening is 1 minus the probability that it will happen.

This video is a guide to probability. Expressing probability as fractions and percentages based on the ratio of the number ways an outcome can happen and the total number of outcomes is explained. Experimental probability and the importance of basing this on a large trial is also covered.

Single Events

Example

There are 6 beads in a bag, 3 are red, 2 are yellow and 1 is blue. What is the probability of picking a yellow?

The probability is the number of yellows in the bag divided by the total number of balls, i.e. 2/6 = 1/3.

Example

There is a bag full of coloured balls, red, blue, green and orange. Balls are picked out and replaced. John did this 1000 times and obtained the following results:

• Number of blue balls picked out: 300
• Number of red balls: 200
• Number of green balls: 450
• Number of orange balls: 50

a) What is the probability of picking a green ball?

For every 1000 balls picked out, 450 are green. Therefore P(green) = 450/1000 = 0.45

b) If there are 100 balls in the bag, how many of them are likely to be green?

The experiment suggests that 450 out of 1000 balls are green. Therefore, out of 100 balls, 45 are green (using ratios).

Multiple Events

Independent and Dependent Events

Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice and consider the probability that both are 6's.

We call two events independent if the outcome of one of the events doesn't affect the outcome of another. For example, if we throw two dice, the probability of getting a 6 on the second die is the same, no matter what we get with the first one- it's still 1/6.

On the other hand, suppose we have a bag containing 2 red and 2 blue balls. If we pick 2 balls out of the bag, the probability that the second is blue depends upon what the colour of the first ball picked was. If the first ball was blue, there will be 1 blue and 2 red balls in the bag when we pick the second ball. So the probability of getting a blue is 1/3. However, if the first ball was red, there will be 1 red and 2 blue balls left so the probability the second ball is blue is 2/3. When the probability of one event depends on another, the events are dependent.

Possibility Spaces

When working out what the probability of two things happening is, a probability/ possibility space can be drawn. For example, if you throw two dice, what is the probability that you will get: a) 8, b) 9, c) either 8 or 9?

a) The black blobs indicate the ways of getting 8 (a 2 and a 6, a 3 and a 5, ...). There are 5 different ways. The probability space shows us that when throwing 2 dice, there are 36 different possibilities (36 squares). With 5 of these possibilities, you will get 8. Therefore P(8) = 5/36 .
b) The red blobs indicate the ways of getting 9. There are four ways, therefore P(9) = 4/36 = 1/9.
c) You will get an 8 or 9 in any of the 'blobbed' squares. There are 9 altogether, so P(8 or 9) = 9/36 = 1/4 .

Probability Trees

Another way of representing 2 or more events is on a probability tree.

Example

There are 3 balls in a bag: red, yellow and blue. One ball is picked out, and not replaced, and then another ball is picked out.

The first ball can be red, yellow or blue. The probability is 1/3 for each of these. If a red ball is picked out, there will be two balls left, a yellow and blue. The probability the second ball will be yellow is 1/2 and the probability the second ball will be blue is 1/2. The same logic can be applied to the cases of when a yellow or blue ball is picked out first.

In this example, the question states that the ball is not replaced. If it was, the probability of picking a red ball (etc.) the second time will be the same as the first (i.e. 1/3).

This video shows examples of using probability trees to work out the overall probability of a series of events are shown. Both independent and conditional probability are covered.

The AND and OR rules (HIGHER TIER)

In the above example, the probability of picking a red first is 1/3 and a yellow second is 1/2. The probability that a red AND then a yellow will be picked is 1/3 × 1/2 = 1/6 (this is shown at the end of the branch). The rule is:

• If two events A and B are independent (this means that one event does not depend on the other), then the probability of both A and B occurring is found by multiplying the probability of A occurring by the probability of B occurring.

The probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. The rule is:

• If we have two events A and B and it isn't possible for both events to occur, then the probability of A or B occuring is the probability of A occurring + the probability of B occurring.

On a probability tree, when moving from left to right we multiply and when moving down we add.

Example

What is the probability of getting a yellow and a red in any order?
This is the same as: what is the probability of getting a yellow AND a red OR a red AND a yellow.
P(yellow and red) = 1/3 × 1/2 = 1/6
P(red and yellow) = 1/3 × 1/2 = 1/6
P(yellow and red or red and yellow) = 1/6 + 1/6 = 1/3