6. (10 points total) A system is compoeed of 4 components, each of which is eith

Question

6. (10 points total) A system is compoeed of 4 components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector (zi,Z2,Z3,T4), where zi s equal to one if component i is working and is equal to 0 if component i is failed. (a) (5 points) How many outcomes are in the sample space of this experiment? (b) (5 points) Suppose that the system will work if components 1 and 2 are both working or if components 3 and 4 are both working, or if components 1,3, and 4 are all working. Let W be the event that the system will work. Specify all the outcomes in W 7. (15 points total) Poker dice is played by simultaneously rolling 5 dice. Show that (a) (2 points)P(one pair) 4630; (b) (2 points) Pítwo pair 2315; (c) (2 points)P no two alike).0926; (d) (2 points)P[three alike .1543; (e) (3 points) P[full house) 0386 (f) (2 points)P(four alike) .0193; (g) (2 points) Pffive alike 0008. 8. (10 points total) A pair of fair dice are rolled (a) (5 points) What is the probability that the second die lands on a value less than the value of the first? (b) (5 points) What is the probability that the second die lands on a value less than or equal to 2/3 the value of the first? 9. (15 points total) The chess clubs of two schools consist of, respectively 10 and 12 players. Six members from each club are randomly chosen to participate in a contest between the two schools. The chosen players from one team are then randomly paired with those from the other team, and each pairing plays a game of chess. Suppose that Rebecca and her sister Elise are on the chess clubs at different schools. What is the probability that (a) (5 points) Exactly one of Rebecca and Elise will be chosen to represent her school. (b) (5 points) Rebecca and Elise will be paired. (c) (5 points) Rebecca and Elise will be chosen to represent their schools but will not play each other. 10. (10 points total) Two dice are thrown n times in succession. (a) (5 points) Compute the probability that a double 3 appears at least once? (b) (5 points) How large need n to be to make this probability at least 1/2

Explanation / Answer

6. a. 2^5 = 32 outcomes possible

b. W = {(1, 1, 0, 0), (1, 1, 1, 0), (1, 1, 0, 1), (1, 1, 1, 1), (0, 0, 1, 1), (1, 0, 1, 1), (0, 1, 1, 1)}

8. a. Total outcomes = 6*6 grid = 36

Outcomes in which second die < first die = no. Of elements in the part below the diagonal of the 6*6 matrix, eg the elements like (2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5) = 15 elements

Therefore probability (second die < first die) = 15/36 = 5/12

b. Second die <= 2/3 of first die

First die | Second die

1 | no possible outcome

2 | 1

3 | 1, 2

4 | 1, 2

5 | 1, 2, 3

6 | 1, 2, 3, 4

Total favorable outcomes = 12

Probability = 12/36 = 1/3

9.a. P(exactly one chosen) = (9C5)*(11C6) + (9C6)*(11C5)

b. P(R & E will be paired) = 1/total pairs possible = 1/(36)

c. P( represent but not pair)= P(represent)*P(not pair)

= (9C5)*(11C5)*(1 - P(pair))

= (9C5)*(11C5)*(1 - 1/36)

10. a. Probability of at least one (3, 3) = 1-P(no (3, 3))

= 1- (11/12)^n

b. 1 - (11/12)^n >= (1/2)

(1/2) >= (11/12)^n

log(1/2) <= n*log(11/12)

n >= [log(0.5)/log(11/12)]

n >= 7.97

Therefore minimum n = 8.

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