MBA Super Notes: Statistics: Introduction to Probability

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MBA Super Notes: Statistics: Introduction to Probability

  1. 1. MBA Super Notes© M S Ahluwalia Sirf Business Version 1.0 Introduction to probability
  2. 2. MBA Super Notes© M S Ahluwalia Sirf Business MBA SUPER NOTES Statistics
  3. 3. MBA Super Notes© M S Ahluwalia Sirf Business Probability 1.
  4. 4. MBA Super Notes© M S Ahluwalia Sirf Business Classical definition 4 1 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
  5. 5. MBA Super Notes© M S Ahluwalia Sirf Business Statistical definition 5 1 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
  6. 6. MBA Super Notes© M S Ahluwalia Sirf Business Modern theory of probability 6 1 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.
  7. 7. MBA Super Notes© M S Ahluwalia Sirf Business Odds 7 1 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
  8. 8. MBA Super Notes© M S Ahluwalia Sirf Business Probability in case of multiple events 2.
  9. 9. MBA Super Notes© M S Ahluwalia Sirf Business Sample space 9 2 • 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 BAB
  10. 10. MBA Super Notes© M S Ahluwalia Sirf Business 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 10 2 Example • For the sample space we’re focusing on: • 𝑃 𝐴 = 𝑛 𝐴 𝑛 = 6 15 • 𝑃 𝐵 = 𝑛 𝐵 𝑛 = 5 15 • 𝐴 ∩ 𝐵 = 𝑃 𝐴𝐵 = 𝑃 𝐵𝐴 = 𝑛 𝐴𝐵 𝑛 = 2 15 • 𝐴 ∪ 𝐵 = 𝑃 𝐴 + 𝐵 = (𝑛 𝐴+𝑛 𝐵−𝑛 𝐴𝐵) 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
  11. 11. MBA Super Notes© M S Ahluwalia Sirf Business 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 11 2 Conditional probability • Probability of happening of B assuming that A has already happened • Symbolically, P(B/A) { 𝑃 𝐵 𝐴 = 𝑛 𝐴𝐵 𝑛 𝐴 = 2 6 } ≥ 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.
  12. 12. MBA Super Notes© M S Ahluwalia Sirf Business 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) 12 2 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
  13. 13. MBA Super Notes© M S Ahluwalia Sirf Business 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 − 𝑛 𝐴 • A and its complement are mutually exclusive events, 𝑃 𝐴 + 𝑃 𝐴 𝑐 = 1 • Example: In our sample space out of 15 points 6 correspond to A so 𝑛 𝐴 𝑐 = 15 − 6 = 9 Relationships between events (2/2) 13 2
  14. 14. MBA Super Notes© M S Ahluwalia Sirf Business Probability theorems 3.
  15. 15. MBA Super Notes© M S Ahluwalia Sirf Business Addition theorem • Also known as theorem of total probability 𝑃 (𝐴 ∪ 𝐵) = 𝑃 𝐴 + 𝐵 = 𝑃 𝐴 + 𝑃 𝐵 − 𝑃 𝐴 ∩ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛 𝐴𝐵 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 15 3 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)
  16. 16. MBA Super Notes© M S Ahluwalia Sirf Business Multiplicatio n 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 Multiplication theorem 16 3 Result related to addition and multiplicatio n theorems 𝑃 𝑎𝑙𝑡𝑒𝑎𝑠𝑡 𝑜𝑛𝑒 = 1 − 𝑃 𝑛𝑜𝑛𝑒
  17. 17. MBA Super Notes© M S Ahluwalia Sirf Business 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 17 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, 𝑃 𝐵𝑖 𝐴 = 𝑃 𝐵𝑖 . 𝑃( 𝐴 𝐵𝑖 ) Σ𝑃 𝐵𝑖 . 𝑃( 𝐴 𝐵𝑖 ) • 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
  18. 18. MBA Super Notes© M S Ahluwalia Sirf Business Probability of a line and an area 4.
  19. 19. MBA Super Notes© M S Ahluwalia Sirf Business 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 19 4 Example • Series of numbers on a dice is a set of discrete variable values.
  20. 20. MBA Super Notes© M S Ahluwalia Sirf Business 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 20 4 f(x) x x+dxx f(x) x ba f(x) x ba f(x) x ba 𝑃 (𝑥 ≤ 𝑎) 𝑃 (𝑥 ≥ 𝑏)𝑃 (𝑎 ≤ 𝑥 ≤ 𝑏)
  21. 21. MBA Super Notes© M S Ahluwalia Sirf Business Expectation 5.
  22. 22. MBA Super Notes© M S Ahluwalia Sirf Business Expectation • The expected value of the outcome • Helps take decisions Expectation 22 5 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 = 𝑝𝑖 𝑥𝑖 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
  23. 23. MBA Super Notes© M S Ahluwalia Sirf Business 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 23 5 Subtraction theorem • Expectation of the difference of two variables is the difference in their expectations. • 𝐸 𝑥 − 𝑦 = 𝐸 𝑥 − 𝐸(𝑦) Multiplicatio n theorem • Expectation of the product of two independent variables is the product of their expectations. • 𝐸(𝑥 × 𝑦) = 𝐸(𝑥) × 𝐸(𝑦)
  24. 24. MBA Super Notes© M S Ahluwalia Sirf Business Other concepts and applications of probability 6.
  25. 25. MBA Super Notes© M S Ahluwalia Sirf Business 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 25 6 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, 𝑅 𝑠 𝑡 = 𝑅1 𝑡 × 𝑅2 𝑡 × … × 𝑅 𝑛(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
  26. 26. MBA Super Notes© M S Ahluwalia Sirf Business 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 26 6 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.
  27. 27. MBA Super Notes© M S Ahluwalia Sirf Business 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 − 1 𝑘2 = 𝑃 𝑥 − 𝑘σ < 𝑥 < 𝑥 + 𝑘σ ≥ 1 − 1 𝑘2 Chebycheff’s lemma 27 6
  28. 28. MBA Super Notes© M S Ahluwalia Sirf Business Do you have any questions or some feedback to share? Send an email to super.msahluwalia@yahoo.com Thank You! 28
  29. 29. MBA Super Notes© M S Ahluwalia Sirf Business M S Ahluwalia, is a visual artist, blogger, designer, photographer, MBA and ex-Banker from India. To know more please follow on Twitter or Facebook: For more Super-Notes: Click Here

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