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Probability Theory

Probability

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. Most probable event occurs most frequently than events with low probability. If the number of repetitions are small, the frequency of event is to a considerable extent a random quality. We conduct an experiment with great number of repetitions and the frequency of the event becomes less and less random and it stabilizes. If the number of independent trials are sufficiently large, we say that frequency has approached the probability of an event. If an experiment is repeated under essentially homogeneous and similar conditions we generally come across two types of situation 

  • The outcome is unique or certain.
  • The result is not unique but may be one of the several possible outcomes.

Random Experiment:

An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. An experiment (or a “trial”) with results that cannot be predicted beforehand, i.e. random.

Sample Space:

The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is donated by S. Each element of Sample space is called a Sample point. A Sample space can be:

  • Discrete

    e.g. S = {1,2,3,4,5,6}

            S = {low, medium, high}

            S={y, n}

  • Continuous

     e.g.  S = R+={x|x>0}

             S={x|10<x<11}

Event:

An Event is a subset of the sample space of a random experiment. An Event either happens or fails to happen as a result of an experiment. It is quite obvious that not all events are equally probable. Some are more probable and some are less probable. Probability theory makes it possible to:

  1. a) Determine the degree of likelihood (probability) of various events
  2. b) To compare them according to their probabilities
  3. c) Predict the outcomes of random phenomenon on the basis of probabilistic estimates.

Example:

If you throw a die (an experiment!), which of the two outcomes is more probable:

A – appearance of six dots

B – appearance of even number of dots

Simple / Elementary Event:

If an event E has only one sample point in a sample space, it is called a simple (or elementary) event. If a Discrete Sample space contains n distinct elements, then there are exactly n simple elements. In a continuous Sample Space there are infinite simple elements.

Example:

  • A vehicle has a Gear Box having 7 Gear positions: 1,2,3,4,5,R,N. A series of Experiments is conducted where the position of Gear is observed every 5 mins during a drive of the vehicle from Point A to Point B.

  Sample Space S = {1,2,3,4,5,R,N}

Each of the observed Gear Positions are Simple Events.

  • During the same drive a separate series of Experiment is conducted where speed of the vehicle is observed every 5 mins. The highest speed possible is 150 kph. 

Sample Space S = {x|0<=x<=150}. 

Each of these observed speeds are Simple Events.

Compound Event:

If an Event has more than one sample point of a sample space, it is called a Compound Event

Example:

In a production line is manufacturing insulated wire ropes of 10m length. The rope is considered as accepted if the measured thickness is within the specification 5+- 0.5 mm, otherwise considered as defective

Experiment: 

A sample of 5 ropes are randomly selected from each production run of 500 ropes and insulation thickness is measured and classified as defective or non-defective.

The following events are considered as Compound event:

  1. a) E=Exactly one of the five ropes found to be defective
  2. b) F=At least four ropes found to be non defective

The Subset associated with these events are:

  1. a) E= {NYYYY, YNYYY, YYNYY, YYYNY, YYYYN}
  2. b) F={NYYYY, YNYYY, YYNYY, YYYNY, YYYYN, YYYYY}

Classical probability:

Let us denote the probability of a random event as P(A)

P(A)  =  ma/n

Where n is the total number of outcomes and ma is the number of outcomes favorable to event A. This is also known as Classical Formula. It is applicable for symmetric experiments which possess symmetry of possible outcomes.

Statistical probability:

  • Most of the experiments in real life are not symmetrical.
  • Hence classical formula cannot be applied to such experiments
  • We find probability in such cases by experimental determination of frequency of the event
  • The frequency of an event in series of N repetitions is the ratio of the number of repetitions, in which the event took place, to the total number of repetitions P(A) = MA/N, where N is total number of repetitions of the experiment and MA is the number of repetitions in which event A occurs

Rules of probability theory:

  1. Probability of an event can attain value between 0 to 1 0<=P(A)<=1
  2. P(not A) = 1 – P(A)
  3. Probability Summation Rule:
  • The probability that one of the two (or several) mutually exclusive (disjoint) events occurs is equal to sum of the probabilities of these events P(A or B) = P(A) + P(B)
  • The General Addition Rule (for non-disjoint events):

P(A or B) = P(A) + P(B) – P(A and B)

Event A = even no of dots in a throw of die

Event B = no of dots >3

P(A or B) = 1/2 + 1/2 – (1/2 * 2/3) = 2/3

Probability Multiplicative Rule:

  • The probability of the combination of two events (sequentially or simultaneously) is equal to the probability of one of them multiplied by the probability of the other provided that the first one has occurred P(A and B) = P(A). P(B/A) where P(B/A) is called conditional probability of event B calculated for the condition that event A has occurred
  • For independent events, P(B/A) = P(B)
  • Two events A and B are called independent if the fact that one event has occurred does not affect the probability that the other event will occur.
  • In such cases: P(A and B) = P(A). P(B)
  • The idea of disjoint events is about whether or not it is possible for the events to occur at the same time.
  • The idea of independent events is about whether or not the events affect each other in the sense that the occurrence of one event affects the probability of the occurrence of the other.
  • If the events are disjoint, then they cannot be independent. A and B disjoint implies that if event A occurs then B does not and vice versa. Knowing that event A has occurred dramatically changes the likelihood that event B occurs – that likelihood is 0. This implies that A and B are not independent.

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