Problem 1: (5 Points) A maintenance firm has gathered the following information regarding the failure mechanisms for air conditioning systems Electrical problemGas leak Yes NoTotal Yes 56 24 80 No 14 6 20 Total 70 30 100If this is a representative sample of AC failure, find the probability that:a) (1 Point) The failure involves a gas leak.b) (1 Point) There is an electrical problem given that there was a gas leak.c) (1 Point) There is at least one of the failures.d) (1 Point) Does gas leak depend on electrical failure? Explain?e) (1 Point) Are the gas leak and electrical failure mutually exclusive? Explain?Problem 2: (6 Points) Joe is considering pursuing an MBA degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 25 percent for University A and 40 percent for University B.a) (1 Point) Is the acceptance decision at University A independent of the acceptance decision at University B? b) (1 Point)c) (1 Point)d) (1 Point)e) (1 Point)f) (1 Point)What is the probability that Joe will be accepted at both universities?What is the probability that Joe will be accepted at University A and rejected at University B? What is the probability that Joe will not be accepted at either university?What is the probability that Joe will be accepted by at least one of the two universities? What is the probability that Joe will be accepted at one, and only one, university?Problem 3: (6 Points) Consider the following probability mass function x -5 2 5 7P(x) 0.2 0.3 0.3 0.2Finda) (1Point)P(X2)c) (1Point) P(X?2)d) (1 Point)P(2?X?7)e) (1.Point) Expected value E(X)f) (1 Point) Standard deviation of X, ?x.Problem 4: (6 Points) A report on high school graduation stated that 85 percent of high school studentsgraduate. Suppose 6 high school students are randomly selected from different schools. 1 a) (1 Point) What is the probability that all of these six students graduate?b) (1 Point) What is the probability that exactly three of the six students graduate?c) (1 Point) What is the probability that at least three of the six students graduate?d) (1.5 Point) What is the probability that between 2 and 4 (inclusive) school students graduate?e) (1.5 Point) Calculate the expected value and standard deviation of the graduates.Problem 5: (5 Points) Use MegaStat Output to solve this problemThirty percent of all customers who enter a store will make a purchase. Suppose that six customers enter the store and that these customers make independent purchase decisions. Let x be the number out of the 6 customers who will make a purchase.Find:a) (1 Point) The probability that exactly five customers make a purchase.b) (1 Point) The probability that at least three customers make a purchase.c) (1 Point) The probability that two or fewer customers make a purchase.d) (1 Point) The probability that at least one customer makes a purchase.e) (1 Point) The mean and the standard deviation of the number of customers who makes a purchaseBinomial distribution6n 0.3X P(X)0 0.117651 0.302532 0.324143 0.185224 0.059545 0.010216 0.000731.00000Problem 6: (5 Points) The price of a stock is uniformly distributed between $30 and $50.a) (1 Point) What is the probability that the stock price will be more than $40?b) (1 Point) What is the probability that the stock price will be less than or equal to $35?c) (1 Point) What is the probability that the stock price will be between $35 and $45?d) (1 Point) Determine the expected price of the stock.e) (1 Point) Determine the standard deviation for the stock priceProblem 7: (7 Points) During the past six months, 73.2 percent of US households purchased sugar. Assume that these expenditures are approximately normally distributed with a mean of $8.22 and a standard deviation of $1.10.a) (1 Point) Find the probability that a household spent less than $5.00 on sugar.b) (1 Point) Find the probability that a household spent more than $10.00 on sugar.c) (1 Point) What proportion of the households spent between $5.00 and $9.00 on sugar?d) (1 Point) 99 percent of the households spent less than what amount?pcumulative probability 0.11765 0.42018 0.74431 0.92953 0.98906 0.99927 1.00000 2e) (1 Point) 80 percent of the households spent more than what amount?f) (2 Point) Calculate the IQR of the household expenditure

Problem 3: [E Paints] Consider the following probability mass function Binomial distribution B n 0.3 p M {5 Points) A maintenance rm has gathered the following information raga cumulative anisms for air conditioning systems X PfXJ probabimj.' 0 0.11165 0.11?65 Electrical Total 1 0.30253 0.42010 problem 2 0.32414 0.?4431 3 0.10522 0.92953 Yes 30 4 0.05954 0.93906 5 0.01021 09992? No 20 3 0.00073 1.00000 1.00000 100