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Engineers presenting a three-point single-link robotic system shown in the accompanying figure. Each point (A, B, or C) in the physical space of the system has either an "obstacle" status or a "free\" status. There are two single links in the system: AB and BC. Alink has a "free" C status if and only if both points in the link are "free"; otherwise the link has an "obstacle" status. Of interest is the random variable x: the total number of links in the system that are "free." Complete parts aand b below. A The possible values of x are 0, 1, and 2. b. The researchers stated that the probability of any point in the system having a "free" status is 0.8. Assum the three points in the system operate independently, find the probability distribution for x. p(0) = p(1) = PR2) = (Round to three decimal places as needed.) According to a recent study, 3 in every 8 women has been a victim of domestic abuse. This probability was obtained from a survey of 1,000 adult women. Suppose we randomly sample 25 women and find that 9 have been abused. Complete parts a through below. a. What is the probability of observing 9 or more abused women in a sample of 25 if the proportion p of women who 3 are victims of domestic abuse is really p = E? The probability of observing 9 or more abused women in a sample of 25 is (Round to four decimal places as needed.) A hotel guest satisfaction study revealed that 65% of hotel guests were aware of the hotel's "green\" conservation program. Among these guests, 71% actually participate in the program by reusing towels and bed linens. In a random sample of 14 hotel guests, consider the number (x) of guests who are aware and participate in the hotel's conservation efforts. a. Explain why x is (approximately) a binomial random variable. b. Use the rules of probability to determine the value of p for this binomial experiment. c. Assume p =0.46. Find the probability that at least 9 of the 14 hotel guests are aware of and participate in the hotel's conservation efforts. a. Complete the explanation below. There are 14 identical trials. For each trial, there are two possible outcome(s). The probability of each possible outcome is the same for each trial, and the trials are all independent. b. p= 0.4615 (Round to four decimal places as needed.) c. The probability is | |. (Round to four decimal places as needed.) A study by a reputable research organization found that when presented with prints from the same individual, a fingerprint expert will correctly identify the match 92% of the time. In contrast, a novice will correctly identify the match 77% of the time. Consider a sample of five different pairs of fingerprints, where each pair is a match. Complete parts a and b below. a. What is the probability that an expert will correctly identify the match in all five pairs of fingerprints? The probability is 0.6591 . (Type an integer or decimal rounded to four decimal places as needed.) b. What is the probability that a novice will correctly identify the match in all five pairs of fingerprints? The probability is | |. (Type an integer or decimal rounded to four decimal places as needed.) According to a recent poll, half of all pet owners would get their next dog or cat from a shelter. Consider a random sample of 20 pet owners and define x as the number of pet owners who would acquire their next dog or cat from a shelter. Assume that x is a binomial random variable. Complete parts a through f below. a. For this binomial experiment, define a success. A success is a pet owner who did not acquire their dog or cat from a shelter. A success is a pet owner who acquired their dog or cat from a shelter. A success is a pet owner who would acquire their next dog or cat from a shelter. A success is a pet owner who would not acquire their next dog or cat from a shelter. b. For this binomial experiment, what is n? n= 20 c. For this binomial experiment, what is p? p= 0.5 (Round to one decimal place as needed.) d. Find P(x=15). P(x=15)= 0.015 (Round to three decimal places as needed.) \fA recent poll of over 2,000 adults was designed to answer the question "Are adults superstitious?" One survey iter concerned the phrase "see a penny, pick it up, all day long you'll have good luck.\" The poll found that just two-fifths adults (40%) believe finding and picking up a penny is good luck. Consider a random sample of 10 adults and let x represent the number who believe finding and picking up a penny is good luck. Complete parts a through f below. a. For this experiment, define the event that represents a "success." A success is an adult who believes finding and picking up a penny is bad luck. A success is an adult who believes finding and picking up a penny is good luck. A success is an adult who does not believe finding and picking up a penny is good luck. A success is an adult who does not believe finding and picking up a penny is bad luck. b. Explain why x is (approximately) a binomial random variable. Sampling 2000 adults is equivalent to performing a dice-roll type experiment. The survey contains at least three options, so one of the options can be considered a success while the others are failures. The survey changes from person to person and each survey response is independent of each other. Binomial random variables are always obtained through sampling. The survey contains at least three options, so one of the options can be considered a success while the others are failures. The survey does not change from person to person and each survey response is dependent of all the previous responses. D. Sampling 2000 adults is equivalent to performing a coin-toss type experiment. The survey only contains two options and each option can be considered a success or a failure. The survey does not change from person to person and each survey response is independent of each other. c. Give the value of p for this binomial experiment. P = 0.40 (Type an integer or a decimal. Do not round.) d. Find P(x