Write down the sample space for the following experiments: a. Toss two coins b.

Question

Write down the sample space for the following experiments:

a. Toss two coins

b. Roll two die

c. Pick two students from a list of three {A, B, C}.

2. Find the probabilities of each outcome in the Problem 1.

3. Given the sample space S = {x : 0 x 5} Let A = {x : 0 x 4} and B = {x : 3 x 5} be two events with probabilities P(A) = 0:512 and P(B) = 0:784. Compute P(A U B) and P(A B).

4. Let P(A|B) = 1/2, P(B c ) = 1/3 and P(A B c ) = 1/6. Compute P(A) and P(A c B).

5. A and B are independent events with P(A) = 0:5 and P(B) = 0:2. Compute a) P(A U B) b) P(A c B c ) c) P(A c U B c ).

6. P(A) = 1/2, P(B) = 1/3 and P(A|B) = 1/6. P(B|A) = ?.

7. A medical test correctly identifies a disease 95% of the time, when it is present. While it is correct 99% of the time when it is not present. The disease is known to affect 10% of the population. Suppose a person tests positive in a test. What is the probability that she actually has the disease? What is the probability that a person that tests negative does not have the disease?

Explanation / Answer

(According to Chegg policy, only four sub-questions will be answered. Please post the remaining in another question)

1. a. Toss two coins : S = {HH, HT, TH, TT}

b. Roll two die : S = {(1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6)}

c. Pick two students from a list of three {A, B, C} : S = {AB, BC, AC}

2. Probability of each event in a = 1/4 = 0.25.

Probability of each event in b = 1/36 = 0.0278.

Probability of each event in c = 1/3 = 0.3333.

3. A U B = {x : 0 x 5} = S

=> P(A U B) = P(S) = 1

P(A B) = P(A) + P(B) - P(A U B) = 0.512 + 0.784 - 1 = 0.2960.

4. P(A|B) = 1/2, P(B') = 1/3 and P(A B') = 1/6

P(B) = 1 - 1/3 = 2/3

P(A B) = P(A|B) P(B) = 1/2 * 2/3 = 1/3

P(A) = P(A B) + P(A B') = 1/3 + 1/6 = 1/2

P(A' B) = P(B) - P(A B) = 2/3 - 1/3 = 1/3.

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