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# PROBABILITY

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### PROBABILITY

1. 1. Multiplication Rule• The joint probability of any two events A and B is• P (A and B) = P (A) P (B|A)Or• P (A and B) = P (B) P (A|B)
2. 2. Example• A graduate statistics course has seven male and three female students. The professor wants to select two students at random to help her conduct a research project. What is the probability that the two students chosen are female?
3. 3.  Let A represent the event that the first student chosen is female and B represent the event that the second student chosen is also female. Because there are three female students in a class of ten, the probability that the first student chosen is female is P(A)= 3/10
4. 4.  After the first student is chosen, there are only nine students left. Given that the first student chosen was female, there are only two female students left. It follows that P(B/A)= 2/9
5. 5.  Multiplication Rule for Independent Events The joint probability of any two independent events A and B is P (A and B) = P (A) × P (B)
6. 6. Example: The professor who teaches the course is suffering from the flu and will be unavailable for two classes. The professors replacement will teach the next two classes. His style is to select one student at random and pick on him or her to answer questions during that class. What is the probability that the two students chosen are female?
7. 7. Random variables A random variable is a variable that assumes numerical values associated with events of an experiment Example 1 Observe 100 babies to be born in a clinic. The number of boys, which have been born, is a random variable. It may take values from 0 to 100. Example 2 Number of patients of a clinic daily is a random variable.ent.
8. 8.  Suppose two fair coins are tossed. Here, the sample space is 5 = {TT, TH, HT, HH} Suppose to each of the four sample points in this sample space, a number is assigned as follows. Sample point TT TH HT HH Number 0 1 1 2 Here, the assigned numbers indicate the number of heads obtained in each case. Let the number of heads be denoted by X. Then, X is a function on the sample space. It takes the values 0,1 and 2 with probabilities — P[X=0] = P[no head] = 1/4 P[X=1] = P[one head] = 1/2 P[X=2] = P[two head] = ¼ Here, X is called Random variable or Variate.