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Math 103 05 Probability

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Notation P - denotes a probability Math 103 Statistics and Probability Probability A, B, ... - denote specific events P (A) - denotes the probability of event A occurring P (A and B) – denotes the probability of event A and event B occuring simultaneously P (A or B) – denotes the probability that event A occurs or event B occurs (or both). CJD Classical Probability Method 1: Classical Approach If a procedure has n different simple events, each with an equal chance of occurring, and s is the
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  1 Math 103Statistics andProbability Probability CJD Notation P - denotes a probabilityA, B, ... - denote specific eventsP (A) - denotes the probabilityof event A occurringP (A and B) – denotes the probabilityof event A and event B occuringsimultaneouslyP (A or B) – denotes the probability thatevent A occurs or event B occurs(or both). CJD Classical Probability Method 1: Classical Approach If a procedure has n  different simple events, each with an equalchance of occurring, and s  is the number of ways event A canoccur, then P(A) = number of ways A can occurnumber of differentsimple events sn = Equally Likely Simple EventsIf there are n  simple events in the sample space andthey are all equally likely, then the probability of theoccurrence of each one is 1/ n  CJD Empirical Probability Method 2: Experimental Approach Conduct (or observe) an experiment a large number of times, and count thenumber of times event A actually occurs, then an estimate of P(A) is P(A) = number of times A occurrednumber of times trial was repeatedLaw of Large Numbers: As a procedure is repeated again and again, theexperimental probability of an event tends toapproach the actual probability. Error is about n 1  2 CJD Classical vs Empirical The classical approach is the actual probability.The relative frequency approach is an approximation. CJD Subjective Probability Method 3: Subjective Approach P(A), the probability of A, is found by simply guessing or estimatingits value based on knowledge of the relevant circumstances. CJD Probability Limits ã The probability of an impossible event is 0 .ã The probability of an event certain to occur is 1 .ã 0 ≤≤≤≤ P(A) ≤≤≤≤ 1 CertainLikely50-50 ChanceUnlikelyImpossible10.50 ãThe total of theprobabilities of allsample points in asample space must equal 1 CJD Computing Probabilities Step 1 : Determine the sample space in the problem. Step 2 : Determine the sample pointsfor the event being considered Step 3 : Assign probabilities to the sample points Step 4 : Determine applicable probability formulasto compute the desired probabilityIf the event is complicated, break it down intoseveral “simpler” events.  3 CJD Example 960,598,2 552 =    What is the probability that a 5-card hand from a standard52-card deck is a flush ?Number of ways to have a 5-card hand =Number of ways to have a flush = 148,5287,14 5134 =⋅=    ⋅ Probability of a flush = 5148/2598960 = 0.00198What is the probability that a 5-card hand from a standard52-card deck is a royal flush ?Probability of a royal flush = 4/2598960 = 0.00000154 CJD Complementary Events P(A) The complement of event A, denoted by A,consists of all outcomes in which event A does not occur. P(A) Note: Sometimes, instead of A, the notation A’ is used. P(A) + P(A’) = 1   CJD Example Example: Testing CorollasToyota wants to conduct a test of its new Corolla model. A pool of 50drivers has been recruited, 20 or whom are men. When the firstperson is selected from this pool, (a) what is the probability of getting a man? (b) What is the probability of getting a woman?P(selecting a man) = num of men drivers / num of drivers= 20 / 50 = 0.4P(selecting a woman) = num of women drivers / num of drivers= 30 / 50 = 0.6OR because selecting a man and selecting a woman arecomplementary events,P(selecting a woman) = 1 – 0.4 = 0.6   CJD Another Example Find the probability that a couple with 3 children has at least 1girl. Assume a child is equally likely to be a boy as to be a girl. If P(A) = P(getting at least 1 girl), thenP(A) = 1 - P(A’)where P(A’) is P(getting no girls)P(A’) = (0.5)(0.5)(0.5) = 0.125P(A) = 1 - 0.125 = 0.875  4 CJD Odds  Odds in favor of event A is thereciprocal of the odds against that event,b:a (or ‘b to a’)  Odds against event A occurring is theratio P(A’) / P(A), usuallyexpressed in the form of a:b(or ‘a to b’), where a  and b  areintegers with no common factors   CJD Odds Example Example: Testing CorollasToyota wants to conduct a test of its new Corollamodel. A pool of 50 drivers has been recruited, 20or whom are men. ã The odds against selecting a man is 3:2ã The odds in favor of selecting a man is 2:3 CJD Probability of Compound Events In general, P(A or B) = P(A) + P(B) - P(A and B) If A and B are mutually exclusive events,Then P(A and B) = 0So, P(A or B) = P(A) + P(B) CJD Men Women Boys Girls TotalsSurvived 332 318 29 27 706Died 1360 104 35 18 1517Total 1692 422 64 56 2223 Example Find the probability of randomly selecting a man or a boy.P(man or boy) = 1692 + 64 = 1756 = 0.7902223 2223 2223 * Mutually Exclusive * Survival data after a plague on a small town :

Lecture1-SSP-2007

Nov 24, 2017

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Nov 24, 2017
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