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# Probability Problems

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I need the full detailed solution of the following 6 probability questions.

1) Given S = {a, b, c, d, e, f, g, h, i, j, k, l}, A = {a, b, c, d, e, f, g, h}, and B = {g, h, k, l}, list the elements of [(A C ∪ B ) C ∪ B C] C.

2) For S = {1, 2, 3, 4}, A = {2, 3, 4} ∈A, and B = {3, 4} ∈ A, where A is an algebra, list the required elements of A.

3) An experiment is performed by rolling a single fair, six-sided die three times. The events A1, A2, and A3 are defined as follows: Ai = {outcome &#8804; 4 on the ith roll} for i = 1, 2, 3. Determine P(A1 ∪ A2 ∪ A3).

4) A letter is chosen at random from the word ELECTRICAL and a letter is chosen at random from the word COMPUTER. In each word, all letters are equally likely to be chosen. Determine the probability that the same letter will be chosen from each word.

5) A company owns 250 multimeters, including 150 which are new, 30 which are five years old, and 70 which are ten years old. Two of the new meters are defective, fifteen of the five-year-old meters are defective, and sixty of the ten-year old meters are defective. A meter is selected at random and found to be defective. What is the probability that it is five years old?

6) Two different programs are used to detect malware on a computer. Program A has a 60% chance of detecting a randomly chosen virus, and program B has a 50% chance of detecting a virus. The probability that both programs detect a given virus is 0.2. Determine the probability that neither program will detect a given virus, and determine whether detection by program A and detection by program B are statistically independent events.

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1) Given S = {a, b, c, d, e, f, g, h, i, j, k, l}, A = {a, b, c, d, e, f, g, h}, and B = {g, h, k, l}, list the elements of [(AC ∪ B)C ∩ BC]C.
S = {a, b, c, d, e, f, g, h, i, j, k, l}
A = {a, b, c, d, e, f, g, h}
AC = {i, j, k, l}
B = {g, h, k, l}
BC = {a, b, c, d, e, f, i, j}
AC ∪ B = {g, h, i, j, k, l}
(AC ∪ B)C = {a, b, c, d, e, f}
(AC ∪ B)C ∩ BC = {a, b, c, d, e, f}
[(AC ∪ B)C ∩ BC]C = {g, h, i, j, k, l}

2) For S = {1, 2, 3, 4}, A = {2, 3, 4} ∈ A, and B = {3, 4} ∈ A, where A is an algebra, list the required elements of A.

The required elements of A = {φ, S, A, B, AC, BC, AUB}, where

S = {1, 2, 3, 4}

A = {2, 3, 4}

B = {3, 4}

AC = {1}

BC = {1, 2}

AUB = {2, 3, 4}

3) An experiment is performed by rolling a single fair, six-sided die three times. The events A1, A2, and A3 are defined as follows: Ai = {outcome ≤ 4 on the ith roll} for i = 1, 2, 3. Determine P(A1 ∪ A2 ∪ A3).

Given that Ai = {outcome ≤ 4 on the ith roll} for i = 1, 2, 3.
Therefore,

A1 = {1, 2, 3, 4}

A2 = {1, 2, 3, 4}

A3 = {1, 2, 3, 4}

Now, A1UA2UA3 = {1, 2, 3, 4}

We know that the sample space, S = {1, 2, 3, 4, 5, 6}

Therefore, P (A1 ∪ A2 ∪ A3) =

= n (A1 ∪ A2 ∪ A3)/n (S)

= 4/6

= 2/3

= 0.6667

4) A letter is chosen at random from the word ELECTRICAL and a letter is chosen at random from the word COMPUTER. In each word, all letters are equally likely to be chosen. Determine the probability that the same letter will be chosen from each word.