Probability

Probability: Sample Space and Events

• Experiment: any process that generates a set of data.

• numerical data, representing counts or measurements,

• or categorical data, which can be classified according to some criterion

• The set of all possible outcomes of a statistical experiment is called the samplespace and is represented by S.

• An event is a subset of a sample space

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• The fundamental principle of counting, often referred to as the multiplication rule

• A permutation is an arrangement of all or part of a set of objects.

permutation vs combination

Probability of an Event

• The likelihood of the occurrence of an event

• resulting from such a statistical experiment

• Evaluated by means of a set of real numbers,

• called weights or probabilities, ranging from 0 to 1.

• if a coin is not balanced, we could estimate the probabilities of heads

and tails by tossing the coin a large number of times and recording the outcomes.

• According to the relative frequency definition of probability, the true probabilities would be the fractions of heads and tails that occur in the long run.

• Another intuitive way is the indifference approach.

• if you have a die that you believe is balanced, you determine that the probability that each of the six sides up after a throw is 1/6.

• The use of intuition, personal beliefs, and other indirect information atprobabilities is referred to as the subjective definition of probability.

• foundation is the statistical experiment rather than subjectivity, viewed as the limiting relative frequency.

Conditional Probability

• The probability of an event B occurring when it is known that some event A has occurred is called a conditional probability and is denoted by P(B|A).

• The symbol P(B|A) is usually read “the probability that B occurs given that A occurs” or simply “the probability of B, given A.”

Independent Events

EXAMPLE If the probability that A will be alive in 20 years is 0.7 and the probability that B will be alive in 20 years is 0.5, then the probability that they will both be alive in 20 years is ? • 0.7 x 0.5 = 0.35

The Product Rule, or the Multiplicative Rule

If the probability that A will be alive in 20 years is 0.7 and the probability that B will be alive in 20 years is 0.5, then the probability that they will both be alive in 20 years is: 0.7 x 0.5 = 0.35

Bayes’ Rule

Bayesian statistics:• A collection of tools that is used in a special form of

statistical inference • applies in the analysis of experimental data• in practical situations in science and engineering. • one of the most important rules in probability theory.

Denote by A, B, and C the events that a grand prize is behind doors A, B,and C, respectively. Suppose you randomly picked a door, say A. Thegame host opened a door, say B, and showed there was no prize behindit. Now the host offers you the option of either staying at the door thatyou picked (A) or switching to the remaining unopened door (C). Useprobability to explain whether you should switch or not.

Bayes’ Theorem to Solve Monty Hall Problem