Question

Stat 352 Homework Set 2 Fall 2019: Conditional Probability and Independence Deadline: Monday November 11, 2019 (1) In throwing two dice with the sample space Define the following events on : = {(x,y):x, y = 1,2,3,4,5,6). A = {sum less than 4) = {(x, y): x + y < 4, x, y = 1,2,3,4,5,6) B = {first number is 1) = {(x,y): x = 1, y = 1,2,3,4,5,6) C = {sum of number is 5) = {(x,y): x + y = 5, x, y = 1,2,3,4,5,6} D = (numbrs are not the same} = {(x, y): x y , x, y = 1,2,3,4,5,6} E = (Numbers are the same} = ((x,y): x = y, x, y = 1,2,3,4,5,6) Calculate: P(E), (ii) P(). (it) P( A B) (iv) PC:D),P(AB)? (2) In throwing two dice with the sample space 2 = {(x, y):x, y = 1,2,3,4,5,6). What is the probability of a sum of 5 if they land on different numbers? In medical diagnosis, we have the following medical data: P(Heart disease )=P(H)=0.40 P(Diabets)=P(D)=0.35 P(Liver disease)=P(L)=0.25 If these diseases (H, D, L) show symptoms ( A ) with the following conditional probabilities: P(AH) = 0.80, P(AD) = 0.60P (AL) = 0.90 Calculate: P(AH), P(A), and P(HA) ? are A and H independent? (3) A fair die is rolled twice, independently. (a) Consider the events: A = "the first number that show up is a 6" B = "the sum of the two numbers obtained is equal to 7" C=" the sum of the two numbers obtained is equal to 7 or 11" (i) Calculate P(BIC) (ii) Calculate P(AB) (iii) Are A and B independent events? (b) Considering rolling two dice. Knowing that an even number was obtained on the first roll, what is the probability that the sum of the two numbers obtained is equal to 4? (a) Suppose that P(A) = P(B) = and that P(AIB) = P(B). Calculate P(ANB) (b) Suppose that P(ANB) = P(A|B') = and P(B) = Calculate P(A)

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Answer 1

2) There are possible outcomes of ordered pair in two tosses of a six sided die.

The sum 5 can occur in

1+4=5

2+3=5

3+2=5

4+1=5

So there are 4 ways. Thus,

The events are disjoint and exhaustive. Using total probability theorem,

Using conditional probability,

Since , events A and H are not independent.

We are required to solve only one question. Please post the remaining questions as another post.