These are five basic questions about Probability and Statistics,can you help me

Question

These are five basic questions about Probability and Statistics,can you help me to explain and solve them step by step if possible? Anyway your helpful answers will be greatly appreciated!

Problems:

7. Three numbers are randomly selected once from 0 to 9. Find the probabilities of the following two events: Al three numbers don't contain 0 and 5, A2 three numbers don't contain 0 or 5 8. 6 students live a dorm, find the probabilities of the following events (1) The birthday of at least one student is in October; (2) Among 6 students, the birthdays of 4 students are exactly in October: (3) Among 6 students, the birthdays of 4 students are exactly in a same month. 9. A batch of products contain products A, B, and C. Suppose that products A, B, and C account for 60 30 and 10% of total products, respectively. Now, a product that randomly withdrawn is not product C, find the probability that this product is product A 10. A dorm is equipped with two warning systems. The working probabilities of systems 1 and 2 are 0.92 and 0.93, respectively. Under the condition that system 1 is down, the probability that system 2 is working is 0.85. Find the probability that (1) systems 1 and 2 are working: (2) system 2 is down and system 1 is working: (3) system 1 is working given that system 2 is down. 11. Let 0 POA)

Explanation / Answer

(a) A1 = Numbers don't contain 0 or 5

n(S) = 10C3 = 120

n(A1) = 8C3 = 56

P(A1) = 56/120 = 7/15

(b) A2 = Numbers contain 0 or 5

n(S) = 120

n(A2) = 8C1 = 8

P(A2) = 8/120 = 1/15

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