mba super notes: statistics: introduction to probability

MBA Super Notes © M S Ahluwalia Sirf Business

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Introduction to probability

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MBA SUPER NOTES

Statistics





Probability

1.





Classical definition

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Definition • If an event can happen in m ways out of a total of n possible equally likely ways, the probability of its happening is defined as:

𝑃 𝐸 =𝑚

𝑛

• The probability of an event is a fraction between 0 and 1, or in terms of percentages, it is between 0 and 100.

• If the event is certain to happen, its probability is 1 or 100%

• If the event cannot occur i.e. it is impossible to happen, its probability is 0 or 0%.

Example • If dice is thrown, there are six equally likely possibilities – 1, 2, 3, 4, 5 and 6. The probability occurrence of each possibility is 1/6





Statistical definition

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Definition • The relative frequency of occurrence of the event when the number of observations is very large

• Also known as Empirical or Frequency definition

Example • In a class of 100 students, there are 60 boys and 40 girls

• The probability that a student selected at random from the class will be a boy is 60/100 = 3 / 5

Application Buffon’s Needle problem

• If parallel lines at equal distance, say d, are drawn on a plane surface and a needle/stick of length d/2 is thrown on this surface, then the probability that it would intersect any line is 1 𝜋 .

• This information can be used to evaluate the value of π by the experiment of throwing a large number of small sticks, and observing the number of sticks intersecting the lines. If the number of sticks thrown are hundred, then:

π = 1 𝑛

where,

n = number of needles/sticks intersecting the lines





Modern theory of probability

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Random experiment

• An experiment whose outcome cannot be predicted with complete certainty

Sample point

• Outcomes of an experiment

Sample space

• Composed of set of points representing all possible outcomes

• Two types:

• Discrete: Sample space that consists of finite/countable number of sample points

• Continuous: Sample space that is not discrete

Probability • The probability of an event “E” is defined as:

P E =n(E)

N

where,

n(E) = number of points corresponding to the happening of the event N = total number of points in the sample space.





Odds

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Odds • The ratio of probability of happening of an event to probability of not happening of the event

Formula • If p is probability that an event will occur, and q that it won’t; the odds in favor of the event happening are p/q.

• q = 1 – p

• Odds against the event will be q/p

Example • The probability of getting number 3 on the top face when a dice is thrown is 1/6 , and the probability of not getting number 3 is 5/6.

• Odds in favor of number 6 in the throw of a dice are 1/6 : 5/6 i.e. 1: 5

• Odds against the number 6 are 5/6 : 1/6 i.e. 5 : 1





Probability in case of multiple

events

2.





Sample space

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• The Venn diagram shows all possible unions and intersections of all events

• A and B are the two events in this case

• We’ll use the following information from the diagram in rest of this section:

• n: number of points in the sample space (15)

• nA: number of points corresponding to happening of A (6)

• nB: number of points corresponding to happening of B (5)

• nAB: number of points corresponding to happening of both A and B i.e. event AB (2)

A B AB





Union • 𝐴 ∪ 𝐵

• Read as A union B or, symbolically, as (A + B)

• Refers to the probability of happening of the two events, whether together or not (includes A, B and AB)

Joint probability: Union and intersection

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Example • For the sample space we’re focusing on:

• 𝑃 𝐴 = 𝑛𝐴 𝑛 = 6 15

• 𝑃 𝐵 = 𝑛𝐵 𝑛 = 5 15

• 𝐴 ∩ 𝐵 = 𝑃 𝐴𝐵 = 𝑃 𝐵𝐴 = 𝑛𝐴𝐵 𝑛 = 2 15

• 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝐵 = (𝑛𝐴+𝑛𝐵−𝑛𝐴𝐵) 𝑛 = 9 15

Intersection • 𝐴 ∩ 𝐵

• Read as A intersection B or, symbolically, as (AB*)

• Refers to the probability of happening of both events together (AB)

*A and B are simple events, whereas, AB is a compound event





Marginal probability

• The probability of happening of an event, say A, independent of the happening of the other events in the sample space (in this case B)

• Marginal probability of A = P(A) = P(AB) + P(ABc)*

• It is called so because it leads to the marginalization of the events, other than the one in focus

Marginal and conditional probability

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Conditional probability

• Probability of happening of B assuming that A has already happened

• Symbolically, P(B/A)

{ 𝑃𝐵

𝐴=𝑛𝐴𝐵𝑛𝐴 = 26 } ≥ 𝑃(𝐵)

• Explanation: If event A has already happened, the sample space for B is reduced to nA . And, number of possible occurrences of B are reduced to nAB.

*Probability of happening of A when B does not happen. ‘c’ stands for complement.





Independent events

• If the occurrence or non-occurrence of event A does not affect the occurrence or non-occurrence of event B , then P(B/A) is same as P(B), and A and B are said to be independent events.

• Example: Probability of getting 6 on the second throw of a dice is independent of the number which appeared on the first throw of the dice

• Test: Probability of joint happening of A and B must be equal to the products of their individual probabilities, i.e.,

𝑃 𝐴𝐵 = 𝑃 𝐴 × 𝑃(𝐵)

Relationships between events (1/2)

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Dependent events

• Events which are not independent

Mutually exclusive events

• Events which do not happen together

𝑃 𝐴 ∩ 𝐵 = 0

• Example: In a single throw of single dice 1 and 3 (or any other number) cannot occur together

• If A and B are mutually exclusive, 𝑃 𝐵 𝐴 = 𝑃(𝐵)

• Mutually exclusive events are not independent, as the happening of one event should guarantee not happening of the other, and vice-a-versa





Complement • Complement of an event refers to all the points in sample space corresponding to not happening of the event

• Notation: Put a superscript ‘c’ next to the event

• Complement of A = Ac = n − 𝑛𝐴

• A and its complement are mutually exclusive events, 𝑃 𝐴 + 𝑃 𝐴𝑐 = 1

• Example: In our sample space out of 15 points 6 correspond to A so 𝑛𝐴𝑐 = 15 − 6 = 9

Relationships between events (2/2)

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Probability theorems

3.





Addition theorem

• Also known as theorem of total probability

𝑃 (𝐴 ∪ 𝐵) = 𝑃 𝐴 + 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵 =𝑛𝐴 + 𝑛𝐵 − 𝑛𝐴𝐵

𝑛

• For more than 2 events:

𝑃 𝐴 ∪ 𝐵 ∪ 𝐶= 𝑃 𝐴 + 𝑃 𝐵 + 𝑃 𝐶 − 𝑃 𝐴 ∩ 𝐵 − 𝑃 𝐵 ∩ 𝐶 − 𝑃 𝐶 ∩ 𝐴+ 𝑃 𝐴 ∩ 𝐵 ∩ 𝐶

• For mutually exclusive event intersections are equal to 0, therefore,

• 𝑃 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝑃 𝐵

• 𝑃 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑃 𝐴 + 𝑃 𝐵 + 𝑃 𝐶

Addition theorem

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Law of total probability

• Special case of theorem of total probability

• When two events are mutually exclusive and exhaustive, A will happen either with B or with the complement of B, therefore, P(A) = P(AB) + P(ABc)





Multiplication theorem

• Also known as theorem of compound probability

𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 . 𝑃(𝐵 𝐴 )

• For independent events:

𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 . 𝑃 𝐵


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Result related to addition and multiplication theorems

𝑃 𝑎𝑙𝑡𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 = 1 − 𝑃 𝑛𝑜𝑛𝑒





A priori and Posterior probabilities

• A priori:

• Situation(cause) is known.

• Probability of happening of an event to be found

• Posterior:

• Event has happened.

• Probability of factor causing the event to be found

Baye’s theorem

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3

Baye’s theorem

• Also known as Baye’s formula.

• Applicable for Posterior probabilities

• Formula: Let there be an event A which can happen only if one of the n mutually exclusive events B1, B2,…. Bn has happened. Then,

𝑃𝐵𝑖𝐴=𝑃 𝐵𝑖 . 𝑃(

𝐴𝐵𝑖)

Σ𝑃 𝐵𝑖 . 𝑃(𝐴𝐵𝑖)

• The formula tells us the probability of each of the Bi causing the event

• P(Bi/A)s (posterior probabilities) indicate the change in P(Bi)s (a priori probabilities) based on the information that A has occurred





Probability of a line and an area

4.





Probability function

• A function, say f(x) which can be used to determine the value of a discrete variable, say x, by substituting x with its value in f(x).

Discrete variable

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Example • Series of numbers on a dice is a set of discrete variable values.





P (x=a)=0, because there are infinite points from a to b, and if each is assigned value >0, then total probability will be >1.

Probability function

• Probability that x takes a value from x to x + dx is given by f(x).dx. Graphically, it is represented as the area f(x).dx shown below :

• Symbolically, P ( x ≤ x ≤ x+dx) = f(x).dx

• f(x) = frequency function = probability density function

Continuous variable

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f(x)

x x+dx x

f(x)

x b a

f(x)

x b a

f(x)

x b a

𝑃 (𝑥 ≤ 𝑎) 𝑃 (𝑥 ≥ 𝑏) 𝑃 (𝑎 ≤ 𝑥 ≤ 𝑏)





Expectation

5.





Expectation • The expected value of the outcome

• Helps take decisions

Expectation

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Expectation for a discrete variable

• If random variable x takes values x1, x2… xi... xn with probabilities p1, p2… pi… pn, then expected value

𝐸 𝑥 = 𝑝𝑖𝑥𝑖

Expectation for a continuous variable

• A continuous variable takes all possible values in a given range. If f(x) is the probability or frequency distribution of the variable x, then expected value

𝐸 𝑥 = 𝑓 𝑥 . 𝑑𝑥





Addition theorem

• Expectation of the sum of two variables is the sum of their expectations.

• 𝐸 𝑥 + 𝑦 = 𝐸 𝑥 + 𝐸(𝑦)

Theorems of Expectation

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Subtraction theorem

• Expectation of the difference of two variables is the difference in their expectations.

• 𝐸 𝑥 − 𝑦 = 𝐸 𝑥 − 𝐸(𝑦)


• Expectation of the product of two independent variables is the product of their expectations.

• 𝐸(𝑥 × 𝑦) = 𝐸(𝑥) × 𝐸(𝑦)





Other concepts and applications of

probability

6.





Reliability • Probability of a device performing its purpose adequately for the period of time intended under the operating conditions encountered

Reliability of components and systems

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Components connected in series

• If two components, say ‘A’ and ‘B’, with reliabilities R1(t) and R2(t), are in series, then the system will function only if both of them function.

• Reliability of the system Rs(t) = Probability that both components will survive an operating time t = R1(t) x R2(t)

• If there are n components, 𝑅𝑠 𝑡 = 𝑅1 𝑡 × 𝑅2 𝑡 × … × 𝑅𝑛(𝑡)

Component A Component B

Components connected in parallel

• If the two components are in parallel, then the system will function even if only one of them functions.

• Reliability of the system Rs(t) = Probability that at least one component will survive an operating time t = R1(t) + R2(t) – R1(t) x R2(t)

Component A

Component B





Law of large numbers

• If an event, with probability ‘p’, is observed repeatedly during independent repetitions, the proportion of the observed frequency of that event to the number of repetitions tends towards ‘p’ as the number of repetitions becomes large.

Law of large numbers

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Example • When a dice is rolled, the probability of getting 6 is 1 6 . The law of large numbers implies that if the proportion of times we get 6 to the total number of times the dice is rolled is recorded, it will tend to 1 6 , as the number of times the dice is rolled increases.





Chebycheff’s lemma

• Also known as Chebyshev’s lemma

• Stated with reference to – a set of values of observations in a data or probability of a random variable

• If a random variable has mean m, and standard deviation σ, then

𝑃 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 𝑤𝑖𝑙𝑙 𝑙𝑖𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑘𝜎 𝑜𝑓 𝑖𝑡𝑠 𝑚𝑒𝑎𝑛 = 1 − 1𝑘2

= 𝑃 𝑥 − 𝑘σ < 𝑥 < 𝑥 + 𝑘σ ≥ 1 − 1𝑘2

Chebycheff’s lemma

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