Estadistica-Probabilidad

Objectives:
i. To demonstrate an understanding of basic probability concepts.
ii. To straightforwardly characterize a sequence of probabilistic event.

1. (3 points) An individual is presented with three different glasses of cola, labeled C, D, and P. He is asked to taste all three and then list them in order of preference. Suppose the same cola has actually been put into all threeglasses.

a. What are the simple events in this ranking experiment and what probability would you assign to each one?
CDP, CPD, DCP, DPC, PCD, PDC. All of them have a probability of 1/6
b. What is the probability that C is ranked first?
2/6 = 1/3 [CDP and CPD]
c. What is the probability that C is ranked first and D is ranked last?
1/6. Only one outcome: CPD

2. (4points) Suppose that the proportions of blood phenotypes in a particular population are as follows: Please show how you arrived at these values

|A |B |AB |O |
|0.42 |0.10 |0.04 |0.44 |

a. Assuming that the phenotypes of two randomly selected individuals are independent of one another, what is the probability that both phenotypes are O?

P (0,0) = 0.44 *0.44 = 0.1936

b. What is the probability that the phenotypes of two randomly selected individuals match?

P (A,A) + P (B,B) + P (AB,AB) + P (0,0) = (0.42*0.42) + (0.10* 0.10) + (0.04*0.04) + (0.44*0.44) = 0.3816

c. What is the probability that the phenotype is A given that you already know that this individual is of type A, B, or AB but not O.


P (A)+ P(B)+P(AB) = 0.42+0.10+0.04 =0.56
P(A)/[P(A)+P(B)+P(AB)]=0.42/0.56 = 0.75

d. If we define the set S={A, B, AB}, complete the following formula:

P(A|S) = P (A∩S)/P(S) = P(A∩A,B,AB)/P(A,B,AB)


3. (5 points) At a large university, in the never-ending quest for a satisfactory textbook, the Statistics Department has tried a different text during each of the last three quarters. During the fall quarter, 500 studentsused the text by Professor Mean; during the winter quarter, 300 students used the text by Professor Median; and during the spring quarter, 200 students used the text by Professor Mode. A survey at the end of each quarter showed that 200 students were satisfied with Mean’s book, 150 were satisfied with Median’s book, and 160 were satisfied with Mode’s book. If a student who took statistics duringone of these quarters is selected at random and admits to having been satisfied with the text, is the student most likely to have used the book by Mean, Median, or Mode? Who is the least likely author? (Hint: Draw a tree-diagram and/or use Bayes’ theorem.)

[pic]
Then we have to multiply:
P(Mean, like)= 0.5 * 0.4 =0.2
P (Mean, not like) = 0.3
P(Median, like) =0.15
P (Median, not like) = 0.15P(Mode, like) = 0.16
P (Mode, not like) = 0.04

P(like)=0.2+0.15+0.16 = 0.51
P(Mean|like) = P(Mean,like)/P(like) =0.2/0.51 = 0.392156
P(Median|like) = P(Median,like)/P(like) = 0.15/0.51= 0.2941176
P(Mode|like) = P(Mode,like)/P(like) = 0.16/0.51 = 0.313725

a) The student is more likely to have used Mean’s book
b) the least likely author is Median

4. (3 points) Suppose that 20% of allindividuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following questions.

A possible random variable can be X = individuals that do not have an adverse reaction. It is a binomial distributionwith probability of p(x)=0.8
a. What is the probability that when the experiment terminates, four individuals have not had adverse reactions?

1-Pnbinom (3,1,0.2) = 1- 0.5904 = 0.4096

b. What is the probability that the drug is administered to exactly five individuals?

0.8*0.8*0.8*0.8*0.2 = 0.08192

c. What is the probability that at most four individuals do not have an adverse...