Download PART 3 MODULE 3 CLASSICAL PROBABILITY ... and more Slides Probability and Statistics in PDF only on Docsity!
PART 3 MODULE 3 CLASSICAL PROBABILITY, STATISTICAL PROBABILITY, ODDS
PROBABILITY Classical Let S be the set of all equally likely outcomes to a random experiment. or theoretical definitions: (S is called the Let E be some particular outcome or combination of outcomes to the experiment. (E is called an The probability of E eventsample space .) is denoted P(E). for the experiment.)
!
P(E) = n(E) n(S) = number of outcomes favorable to E number of possible outcomes EXAMPLE 3.3.1 Roll one die and observe the numerical result. Then S = {1, 2, 3, 4, 5, 6}. Let E be the event that the die roll is a number greater than 4. Then E = {5, 6}
!
P(E) = n(E) n(S) = 26 =. EXAMP Referring to the earlier example (from Unit 3 Module 3) concerning the Requirer LE 3.3.2. What is the probability that a randomly selected story will be about Elvis? National EXAMPLE 3.3.2 solution In solving that problem (EXAMBLE 3.3.14) we saw that storylines. Of those 21 possible story lines, 12 were about Elvis. Thus, if one story line is randomly selected or generated, the probability that it is about Elvis is 12/21, or there were 21 possible roughly .571.
EXAMPLE 3.3.3 An office employs se to receive a free lunch with the boss. What is the probability that the selected employee will be a woman? ven women and five men. One employee will be randomly selected
EXAMPLE 3.3.4 An office employs seven women and five men. Two employee selected for drug screening. What is the probability that both employees will be men?s will be randomly
EXAMPLE 3.3.7 Let G be the event that the die roll is "Elephant." Then G = { } So
! If an event is impossible, then its probability is 0.^ P(G)^ =^ n(G)^ n(S)^ =^06 =^0 Probabilities are never less than 0 We have the following scale: For any event E in any experiment ., 0 ≤ P(E) ≤ 1
EXAMPLE 3.3.8 A jar contains a penny, a nickel, a dime randomly selected (without replacement) and their monetary sum is determined. 1. What is the probability that their monetary sum will be 55¢?, a quarter, and a half-dollar. Two coins are A. 1/25 2. What is the probability t B. 1/32 (^) hat the monetary sum will be 48¢?C. 1/9 D. 1/ A. 1/10 B. 1/9 C. 1/32 D. 0
EXAMPLE 3.3.9 What is the probability of winning the Florida Lotto with one ticket?
EMPIRICAL PROBABILITY
Suppose we observe the game for one weekend. Over this period of time, the game is^ P(E)^ =^ number of trials of the experimentnumber of occurrences of E played 582 times, with 32 winners. Based on this data, we find P(E).
! The law of large numbers^ P(E)^ =^58232 "^ .055 is a theorem in statistics that states that as the number of trials of the experiment increases, the observed empirical probability will get closer and closer to the theoretical probability. We can also refer to popula distributed across a population. The proportion of the population satisfying E.tion statistics to infer to probability of a characteristic statistical probability of an event E is the EXAMPLE 3.3.11 For instance (this is authentic data), a recent (19 the Natural Resources Defense Council, revealed that: 40% of bottled water samples were merely tap water.99) study of bottled water, conducted by 30% of bottled water samples were contaminated by such pollutants as arsenic and fecal bacteria. Let E be the event "A randomly selected sample of bottled water is actually tap water." Let F be the event "A randomly selected sample of bottled water is contaminated." Then: P(E) = 40% =. P(F) = 30% =.
EXAMPLE 3.3.12 According to a recent article fr 63% of cowboys suffer from saddle sores,om the New England Journal of Medical Stuff , 52% of cowboys suffer from bowed legs, and 40% suffer from both saddle sores and bowed legs. Let E be the event "A randomly selected cowboy has saddle sores." The Let F be the event "A randomly selected cowboy has bowed legs." Then P(F) = .52n P(E) =. Likewise, P(cowboy has both conditions) =. ODDS Odds are similar to probability, in that they involve a numerical method for describing the likelihoood of an event. Odds are defined differently however.
! Odds are usually stated as a ratio.^ Odds in favor of E^ =^ n(n(E)^ E"^ )^ =^ number of outcomes unfavorable to Enumber of outcomes favorable to E
(^6) (this data is authentic): 79% of Americans know that "Just Do It" is a Nike slogan. - 8: A poll (1999) by the Colonial Williamsburg Foundation revealed the following 47% know that the phrase "Life, Liberty and the Pursuit of Happiness" is found in the Declaration of I 9% know that George Washington was a Revolutionary War general.ndependence.
6. Pursuit of Happiness" is found in the Declaration of Independence? A. 47/79 What is the probability that an American knows that the phrase "Life, Liberty and the B. 47/13 5 C. 47/88 D. 47/ 7. Revolutionary War general? A. .9 What is the probability that an American knows that George Washington was a B. .1 C. .09 D.. 8. sl A. .79ogan? What is the probability that an American does not know that "Just Do It" is a Nike B. .21 C. 7.9 D. 2. 9. is a Nike slogan? A. 79:100 What are the odds in favor of a randomly selected American knowing that "Just Do It" B. 21:100 C. 79:21 D. 21: 10. chosen from the set {1, 2, 3, ... , 19, 20}. What is the probability that the "combination" has no repeated numbers? A. .00015 A "combination" lock has a three B. .75 C. .15-number "combination" D. .855 where the numbers are 11. test in order to get a C clueless on the other four problems. If he just guesses at the other four problems, what is the probability that he will get a score of 100%? Gomer is taking a 25- in the course. He knows the answers to 21 of the questions, but is-question multiple-choice test. He needs to get (For each multiple-choice problem a 100% on this there are four choices.) A. .25 B. .0625 C. .004 D.. ANSWERS TO LINKED EXAMPLES EXAMPLE 3.3.3 EXAMPLE 3.3.4 EXAMPLE 3.3.8 7/12 or .58310/66 or .1521. D 2. D EXAMPLE 3.3.9 1/22,957,
ANSWERS TO PRACTICE E 1. 5. 9. 62/824/78C ≈ ≈ .051 .756 2.6.10. 34/78D D ≈ XERCISES .436 3.7.11. 0 C C 4.8. 1/15B
