Lecture-1-Probability-概率论

1、Probability: Review,1,Definitions,2,Probability refers to the study of randomness and uncertainty In probability theory we can make inferences about a population based on its distribution to quantify the chances, or likelihoods, associated with various outcomes In science and Engineering: The probab

2、ility of a good part being produced The reliability of a new machine (reliabilities are actually probabilities),Definitions,3,Random Process - Any process whose possible results are and its actual results cannot be predicted with certainty in advance Outcome - Each possible result for a random proce

3、ss. Experiment - process by which an observation or measurement is obtained. Sample space - the set of all possible outcomes in an experiment,Definitions,4,Event - any collection (or subset) of outcomes contained in the sample space. Simple Event - an event that cannot be decomposed, one outcome of

4、the experiment or in the sample space. Compound Event - collection of specified outcomes contained in the sample space. Null Event (): An event with no outcomes. Example: Number of days with a high temperature over 150 degrees F.,Examples,Toss a coin. The sample space is H,T. The event of head is H.

5、 Toss a coin twice. The sample space is HH, HT,TH,TT. The event of getting exactly one head is HT, TH. Battery failures. If a new type D flashlight battery has a voltage that is outside certain limits, that battery is characterized as a failure (F); if the battery has a voltage within the prescribed

6、 limits, it is a success (S). Suppose the experiment consists of testing each battery as it comes off n assembly line until we first observe a success. The sample space is S, FS, FFS, FFFS,.,5,6,Example: Experiment - toss a single die Simple events (or outcomes): E1 : observe a 1 E2 : observe a 2 E3

7、 : observe a 3 E4 : observe a 4 E5 : observe a 5 E6 : observe a 6 Compound events: A : observe an odd number B : observe a number greater than or equal to 4,Definitions,7,The Union of events A and B, denoted by A B and read “A or B” is the event consisting of all outcomes that are either in A or in

8、B or in both events The Intersection of events A and B, denoted by A B and read “A and B”, is the event consisting of all outcomes that are in both A and B.,8,Complement - the complement of event A, denoted by A, is the set of all outcomes in the Sample Space that are not contained in A Mutually exc

9、lusive or Disjoint events - those events, e.g., A and B that have no outcomes in common. This means that if one of them occurs, the other cannot,Definitions,Defining Probabilities and Their Properties,9,Basic definition: The probability of any event, say E, P(E) should equal n/N where: n = number of

10、 times we observe the event N = a very large number of trials I know it is confusing, but I am have not started to teach you how to compute P(E) for a specific event E Q. In our die experiment for example, in the long run, what would we expect P(E1) to equal?,Basic axioms of probability: 1.0 P(E) 1

11、for any event, E. 2. P(Ei) = 1 for all simple events in a sample space. 3.For a set of mutually exclusive events (remember two events are said to be mutually exclusive if when one occurs, the other cannot occur.) P(A1 A2 A3 . An) = P(A1) + P(A2) + P(A3). + P(An),11,Properties of probability The prob

12、ability of a (non-simple) event is the sum of all the mutually exclusive simple events contained within Example: A is the event of having an odd number when tossing a die P(A) = P(E1)+P(E3)+P(E5) = 1/6 + 1/6 + 1/6 = 3/6 = The probability of the complement of an event A is P(A) = 1 - P(A) Example: if

13、 P(rain tomorrow) = .4 then P(no rain tomorrow) = .6 or in the die example P(C) = 1 - P(A) = 1-1/2 = If two events, A and B, are mutually exclusive P(A B) = 0 since both of them cannot occur together,Additive Law of Probability,12,The probability of the union (A B) of two events A and B is: P(A B) =

14、 P(A) + P(B) - P(A B) If A and B are mutually exclusive then P(A B) = 0, hence: P(A B) = P(A) + P(B) Note that we are defining the probability of event A or event B or both with addition.,More Probability Properties,The probability of a union of more than two events For any three events A, B, and C:

15、 P(A B C) = P(A) + P(B) + P(C) P(A B) P(A C) P(B C) + P(A B C),14,When I visit the local library, the probability that someone is reading the current issue of Sports Illustrated is .4, the probability that someone is reading Time is .3, and the probability that at least one of these two magazines is

16、 being read by someone is .5. What is the probability that Both of the magazines are being read? Neither of the two is being read? Exactly one is being read?,Example,Counting Techniques,Counting techniques: Product Rule, Permutations, and Combinations,16,These rules eliminate the need for listing ea

17、ch simple event Product Rule: for ordered pairs (selecting from multiple groups): If the first element or object of an ordered pair can be selected in n1 ways, and the second element of the pair can be selected in n2 ways, then the number of pairs is n1 x n2 Example: If you were choosing between 3 u

18、niversities to do your undergraduate work and 4 to do your graduate work, there would be 12 possible ways of choosing them.,17,If there are k elements (or items/objects you have to choose) and n1 choices for the first, n2 for the second, and so on to nk, then the possible ways of selecting them is n

19、1 x n2 x . x nk Example: In selecting an electrical, a plumbing, and a construction contractor for a particular job, if you can choose from 5 different electrical contractors, 4 plumbing contractors, and 3 construction contractors, then there are 5 x 4 x 3 = 60 possible ways to choose them for the j

20、ob.,Product rule,Example,A family has just moved to a new city and requires the services of both an obstetrician and a pediatrician. There are two easily accessible medical clinics, each having two obstetricians and three pediatricians. The family will obtain maximum health insurance benefits by joi

21、ning a clinic and selecting both doctors from that clinic. In how many ways can this be done?,19,Permutations,20,Combinations,21,Combinations,Permutations and Combinations,Proposition Notice that and since there is only one way to choose a set of (all) n elements or of no elements, and since there a

22、re n subsets of size 1.,Example,A real estate agent is showing homes to a prospective buyer. There are ten homes in the desired price range listed in the area. The buyer has time to visit only three of them. In how many ways could the three home be chosen if the order of visiting is considered? In h

23、ow many ways could the three homes be chosen if the order is disregarded? If four of the homes are new and six of them are not, and if the three homes to visit are randomly chosen, what is the probability that all three are new (do you consider the order)?,Conditional Probability,24,Sometimes, the p

24、robability of an event, say A, will change if another event has already occurred Example: in tossing a die, let A: observing an odd number and B: observing a number greater than or equal to four. We know that P(A) = 1/2. However, if B has occurred (that is we tossed a 4, 5, or 6) then the probabilit

25、y of A becomes 1/3. So that P(A/B) = 1/3. The conditional probability of A, given that B has already occurred, is denoted as P(A/B) or P(A|B).,Conditional Probability,Example,Suppose that of all individuals buying a certain digital camera, 60% include an optional memory card in their purchase, 40% i

26、nclude an extra battery, and 30% include both a card and battery. Consider randomly selecting a buyer and let A = memory card purchased B = battery purchased. Given that the selected individual purchased an extra battery, what is the probability that an optional card was also purchased?,Definition of a Partition,27,S,The Law of