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- 8.2 Q6) Hey! Need help with the following, thank you!Q 53. Prove or Disprove: Prove or disprove by counterexample the following statements. Follow these steps: • Write either TRUE' or FALSE' next to each statement. • If True, then prove the statement using the tabular form and show each step with justification. • If False, then provide one clear counterexample and show the statement does not hold. a) Let A and B be two events in a sample space, S. If ACB then A C B. b) Let A and B be two events in a sample space, S. Then, P(AnB) < min{P(A), P(B)}. c) The following is a valid probability density function: -2Q 53. Prove or Disprove: Prove or disprove by counterexample the following statements. Follow these steps: • Write either 'TRUE’ or FALSE’ next to each statement. • If True, then prove the statement using the tabular form and show each step with justification. • If False, then provide one clear counterexample and show the statement does not hold. a) Let A and B be two events in a sample space, S. If A c B then A° C Bº. b) Let A and B be two events in a sample space, S. Then, P(AnB) < min{P(A), P(B)}. c) The following is a valid probability density function: -2 < x < 0 0 < x < 2 2-x f(x) : otherwise.4) Suppose Cal Crouk is a crooked politician. Suppose that, on average, he lies 2 times in a five minute stretch. Suppose Cal has a half hour speech planned, and let X be the number of lies in Cal's speech. a) How is X distributed? b) What is the mean and variance of X c) What is the probability that Cal lies exactly 20 times?The 17 colleges of interest to a high school senior include 7 that are expensive (tuition. more than 20,000 per year), 7 that are far from home (more than 200 miles away), and 3 that are both expensive and far from home. A) If the student decided to select a college that is not expensive and within 200 miles of home, how many selections are possible? ______ B) If the student to attend a college that is not expensive and within 200 miles from home during his first two years of college, and then will transfer to a college that is not expensive but is far from home, how many selections of two colleges are possible?3. Indicate Type 1 and Type 2 errors in the following cases a) Adding water in tooth paste does not prevent cavities b) A patient has been given the wrong treatment c) My friend does not have their birthday today but I will wish her happy birthday. d) My friend has their birthday today but I don't wish her happy birthday. e) Adding water in tooth paste does not prevent cavities f) A patient has been given the wrong treatment g) My friend does not have their birthday today but I will wish her happy birthday. h) My friend has their birthday today but I don't wish her happy birthday.) Bender Electronics buys keyboards for its computers from another company. The keyboards are received in shipments of 100 boxes, each box containing 20 keyboards. The quality control department at Bender Electronics first randomly selects one box from each shipment and then randomly selects 5 keyboards from that box. The shipment is accepted if not more than 1 of the 5 keyboards is defective. The quality control inspector at Bender Electronics selected a box from a recently received shipment of keyboards. Unknown to the inspector, this box contains 6 defective keyboards.(i). what is the probability that this shipment will be accepted?(ii). what is the probability that this shipment will not be accepted?Need fully correct solution for this ASAP.Let X be the sum of the roll of two fair six sided dice .Find P[X=8] X=7 or X =8]Using the pigeonhole principle Assuming no one has more than 1,000,000 hairs on the head of any person and that the population of Manila was 11,621,500 in 2009, there had to be at least _____ people in Manila in 2009 with the same number of hairs on their heads.There were 550 people registered for a conference. There were only three sessions, whichwere held at different times during the day. No session times conflicted with any othersession time. A post-conference survey that the attendees completed showed that:❖ 500 people attended Session A❖ 450 people attended Session B❖ 425 people attended Session C❖ 400 people attended both Sessions A and B❖ 410 people attended both Sessions B and C❖ 415 people attended both Sessions A and C❖ 390 people attended all three sessions A, B, and CFor the following questions, you may create a Venn Diagram at the bottom or showyour calculations to explain your reasoning.a) How many people attended none of the three sessions? Explain. b) How many people attended Session A but not Sessions B or C? Explain. c) How many people attended exactly two sessions? Explain.Spuds-R-Us makes potato chips. A typical size is the 16-ounce bag of chips. A paidsampler went to various stores and picked bags of chips packaged on various days. Shegot 76 bags of chips. The average weight of the bags is 15.85 ounces with a standarddeviation of 1.2 ounces.a) Is there evidence the company is under filling the bags of chips? State the null and alternativehypothesis. Give the test statistic and the p-value. What is your decision at the significance level5%? Write a conclusion.b) Construct and Interpret an 95% confidence interval for the average weight of bags.SEE MORE QUESTIONS