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2021-07-04 06:35:22 Revision Notes on Probability

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A negative binomial experiment is an experiment which consists of x repeated trials in which each trial can result in just two possible outcomes a success or a failure. In addition to this it has the following properties:

The probability of success, denoted by P, is the same on every trial.

The trials are independent, that is, the outcome on one trial does not affect the outcome on other trials.

The experiment continues until r successes are observed, where r is specified in advance.

Let A1, A2, .... An be a set of mutually exclusive and exhaustive events and E be some event which is associated with A1, A2, ...., An. Then probability that E occurs is given by
P (E) = ∑n(i=1) P(Ai)P(E/Ai).

If the set of n events related to a sample space are pair-wise independent, they must be mutually independent, but vice versa is not always true.

In probability problems which require the application of the total probability formula the events A1 and A2 must fulfill the following three conditions:

A1 ∩ A2 = Φ

A1 U A2 = S.

A1 ∈ S and A2 ∈ S.

Conditions for application of Bayes formula in probability questions include:

A1, A2, ......., An of the sample space are exhaustive and mutually exclusive i.e. A1 U A2 U ........... U An = S

and Ai ∩ Aj = ∅, where j, i = 1, 2, ........ n and i ≠ j

Priority events i.e. within the sample space there would exist an event B such that P (B) > 0.

The main aim is to compute a conditional probability of the form P (Ai /B).

We know that at least one of the two sets of the two probabilities are given below:

P(Ai ∩ B) for each Ai

P(Ai) and P(B/Ai) for each Ai

If in a problem some event has already happened and then the probability of another event is to be found, it is an application of Bayes Theorem. To recognize the question in which Bayes’ theorem is to be used, the key word is “is found to be". Open / Comment

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