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Probability

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- Laws of Chance.
- Language of Uncertainty.
“The scientific interpretation of chance

begins when we introduce probability.”

-- David Ruelle

- The notion of chance has existed for centuries.
- Egyptian tombs from around 2000 B.C.
- Card and Board games from 14th century

- Probability Quantifies Uncertainty.
- 0 P(A) 1
- Interpret P(A)=0 and P(A)=1

- Basis of Inferential Statistics

- Let n be the total number of outcomes possible, and assume that all outcomes are equally likely.
- Let m be the number of distinct outcomes that comprise the event A.
- The probability of event A occurring is:
P(A) = m / n

- The classical definition of probability provides the theoretical probability of event A. The theoretical probability is not always calculable.
- Examples:
- In some situations, it is not possible to count all outcomes.
- The outcomes are not equally likely to occur in all situations.

- Examples:

- The empirical probability of an event is the observed relative frequency of occurrence of that event if the experiment is repeated many times.
- The empirical probability converges to the theoretical probability (truth) as the number of repetitions gets large.

- Experiment
- an activity resulting in an uncertain outcome

- Sample Space (S)
- set of all possible outcomes in an experiment

- Event (A)
- set of some of the possible outcomes of an experiment
- Any event is a subset of the sample space
- An event is said to occur if the outcome of the experiment is a member of it.

- set of some of the possible outcomes of an experiment

- P(A) – denotes the probability of event A occurring ( 0 P(A) 1 )
- n(A) – denotes the number of distinct outcomes in event A
- Classical Definition of Probability:

- The complement of event A (denoted A’) contains all elements in the sample space that are not in A.
- A’ occurs when A does not occur.

- Complement Rule:
- Many problems are easier to solve using the complement.

- A discrete probability distribution specifies the probability associated with each possible distinct value of the random variable.
- A probability distribution can be expressed in the form of a graph, table or formula.
- For example: Let X be the number of heads that you get when you flip 2 fair coins.

- A probability function, denoted P(x), assigns probability to each outcome of a discrete random variable X.
- Properties:

- Results from an experiment in which a trial with two possible outcomes is repeated n times.
- Heads/Tails, Yes/No, For/Against, Cure/No Cure

- One outcome is arbitrarily labeled a success and the other a failure

- n independent trials
- Probability of success is p in each trial
(so q=1-p is the probability of failure)

- Let X be the number of success in n trials, then X is a binomial random variable.
- Often, p is defined to be the proportion of the population with a characteristic of interest, and X is the number sampled with that characteristic of interest.

- Probability Function

n!

P(x) = • px•qn-x

(n - x )! x!

Probability of x successes among n trials for any one particular order

Number of

outcomes with exactly x successes among n trials

- The mean, variance and standard deviation of a binomial random variable with n trials and probability of success p: