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Probability & Bayesian Theorem

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- 1. Probability Dr Azmi Mohd Tamil Dept of Community HealthUniversiti Kebangsaan Malaysia
- 2. Sample Spaces4 A sample space is the set of all possible outcomes. However, some sample spaces are better than others.4 Consider the experiment of flipping two coins. It is possible to get 0 heads, 1 head, or 2 heads. Thus, the sample space could be {0, 1, 2}. Another way to look at it is flip { HH, HT, TH, TT }. The second way is better because each event is as equally likely to occur as any other.4 When writing the sample space, it is highly desirable to have events which are equally likely.
- 3. Sample Spaces4 Another example is rolling two dice. The sums are { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }. However, each of these arent equally likely. The only way to get a sum 2 is to roll a 1 on both dice, but you can get a sum of 4 by rolling a 3-1, 2-2, or 3- 1. The following table illustrates a better sample space for the sum obtain when rolling two dice.
- 4. Example Second DieFirst Die 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12
- 5. Classical ProbabilitySum Freq Relative Freq 4 The relative frequency of a frequency distribution is the probability of the event2 1 1/36 occurring. This is only true,3 2 2/36 however, if the events are4 3 3/36 equally likely.5 4 4/36 4 This gives us the formula for6 5 5/36 classical probability. The7 6 6/36 probability of an event occurring8 5 5/36 is the number in the event9 4 4/36 divided by the number in the sample space.10 3 3/3611 2 2/36 4 P(E) = n(E) / n(S)12 1 1/36
- 6. Empirical Probability4 Empirical probability is based on 4 P(E) = f / n observation. The empirical probability of an event is the relative frequency of a frequency distribution based upon observation.
- 7. Probability Rules4 All probabilities are between 0 and 1 inclusive 0 <= P(E) <= 14 The sum of all the probabilities in the sample space is 14 The probability of an event which cannot occur is 0.4 The probability of an event which must occur is 1.4 The probability of an event not occurring is one minus the probability of it occurring. P(E) = 1 - P(E)
- 8. Mutually Exclusive Events4 Two events are mutually exclusive if they cannot occur at the same time.4 If two events are mutually exclusive , then the probability of them both occurring at the same time is 0. Mutually Exclusive : P(A and B) = 04 If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.
- 9. Specific Addition Rule4 Only valid when the events are mutually exclusive. P(A or B) = P(A) + P(B)
- 10. Example 14 Given: P(A) = 0.20, P(B) = 0.70, A and B are mutually exclusive B B Total A 0 0.2 0.2 A 0.7 0.1 0.8 Total 0.7 0.3 1
- 11. Non-Mutually Exclusive Events4 In events which arent mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted.4 General Addition Rule P(A or B) = P(A) + P(B) - P(A and B)
- 12. Example 24 Given P(A) = 0.20, P(B) = 0.70, P(A and B) = 0.15 B B Total A 0.15 0.05 0.2 A 0.55 0.25 0.8 Total 0.7 0.3 1
- 13. Independent Events4 Two events are independent if the occurrence of one does not change the probability of the other occurring.4 An example would be rolling a 2 on a die and flipping a head on a coin. Rolling the 2 does not affect the probability of flipping the head.4 If events are independent, then the probability of them both occurring is the product of the probabilities of each occurring.
- 14. Specific Multiplication Rule4 Only valid for independent events P(A and B) = P(A) * P(B)
- 15. Example 34 P(A)= 0.20, P(B) = 0.70, A and B are independent. B B Total A 0.14 0.06 0.2 A 0.56 0.24 0.8 Total 0.7 0.3 1
- 16. Dependent Events4Ifthe occurrence of one event does affect the probability of the other occurring, then the events are dependent.
- 17. Conditional Probability4The probability of event B occurring that event A has already occurred is read "the probability of B given A" and is written: P(B|A)4 GeneralMultiplication Rule P(A and B) = P(A) * P(B|A)
- 18. Example 44 P(A) = 0.20, P(B) = 0.70, P(B|A) = 0.40 B B Total A 0.08 0.12 0.2 A 0.62 0.18 0.8Total 0.7 0.3 1
- 19. Independence Revisited4 Thefollowing four statements are equivalent 1.A and B are independent events 2.P(A and B) = P(A) * P(B) 3.P(A|B) = P(A) 4.P(B|A) = P(B)
- 20. The question, "Do you smoke?" was asked of 100 people. Results are shown in the table. . Yes No Total Male 19 41 60 Female 12 28 40 Total 31 69 100•What is the probability of a randomly selected individual being a male who smokes? This is just ajoint probability. The number of "Male and Smoke" divided by the total = 19/100 = 0.19•What is the probability of a randomly selected individual being a male? This is the total for maledivided by the total = 60/100 = 0.60. Since no mention is made of smoking or not smoking, itincludes all the cases.•What is the probability of a randomly selected individual smoking? Again, since no mention ismade of gender, this is a marginal probability, the total who smoke divided by the total = 31/100 =0.31.•What is the probability of a randomly selected male smoking? This time, youre told that you havea male - think of stratified sampling. What is the probability that the male smokes? Well, 19 malessmoke out of 60 males, so 19/60 = 0.31666...•What is the probability that a randomly selected smoker is male? This time, youre told that youhave a smoker and asked to find the probability that the smoker is also male. There are 19 malesmokers out of 31 total smokers, so 19/31 = 0.6129 (approx)
- 21. There are three major manufacturing companies that make a product: Aberations, Brochmailians, and Chompielians. Aberations has a 50% market share, and Brochmailians has a 30% market share. 5% of Aberations product is defective, 7% of Brochmailians product is defective, and 10% of Chompieliens product is defective. This information can be placed into a joint probability distribution Company Good Defective Total Aberations 0.50-0.025 = 0.475 0.05(0.50) = 0.025 0.50 Brochmailians 0.30-0.021 = 0.279 0.07(0.30) = 0.021 0.30 Chompieliens 0.20-0.020 = 0.180 0.10(0.20) = 0.020 0.20 Total 0.934 0.066 1.00•What is the probability a randomly selected product is defective?•What is the probability that a defective product came fromBrochmailians?
- 22. The percent of the market share for Chompieliens wasnt given, but since the marginals mustadd to be 1.00, they have a 20% market share.Notice that the 5%, 7%, and 10% defective rates dont go into the table directly. This isbecause they are conditional probabilities and the table is a joint probability table. Thesedefective probabilities are conditional upon which company was given. That is, the 7% is notP(Defective), but P(Defective|Brochmailians). The joint probability P(Defective andBrochmailians) = P(Defective|Brochmailians) * P(Brochmailians).The "good" probabilities can be found by subtraction as shown above, or by multiplicationusing conditional probabilities. If 7% of Brochmailians product is defective, then 93% is good.0.93(0.30)=0.279.•What is the probability a randomly selected product is defective? P(Defective) = 0.066•What is the probability that a defective product came from Brochmailians?P(Brochmailian|Defective) = P(Brochmailian and Defective) / P(Defective) = 0.021/0.066 =7/22 = 0.318 (approx).•Are these events independent? No. If they were, then P(Brochmailians|Defective)=0.318would have to equal the P(Brochmailians)=0.30, but it doesnt. Also, the P(Aberations andDefective)=0.025 would have to be P(Aberations)*P(Defective) = 0.50*0.066=0.033, and itdoesnt.
- 23. Bayes TheoremLets use the same example, but shorten each event to its one letter initial, ie: A, B, C, and D instead of Aberations, Brochmailians, Chompieliens, and Defective. P(D|B) is not a Bayes problem.This is given in the problem. Bayes formula finds the reverse conditional probability P(B|D).It is based that the Given (D) is made of three parts, the part of D in A, the part of D in B, and the part of D in C. P(B and D) P(B|D) = ----------------------------------------------------- P(A and D) + P(B and D) + P(C and D) Inserting the multiplication rule for each of these joint probabilities gives P(D|B)*P(B) P(B|D) = ------------------------------------------------------------ P(D|A)*P(A) + P(D|B)*P(B) + P(D|C)*P(C)

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