Applied Engineering Probability and StatisticsPractice for the Final Exam.1. How many di°erent bit strings of length 12 containa) exactly three ones?b) at most three ones?c) at least three ones?d) an equalnumber of 0s and 1s?2.LetA; B;andBbe sets and suppose thatjAj=jBj=jCj= 100;jA\Bj=60;jA\Cj= 50;jB\Cj= 40;andjA[B[Cj= 175:How many elements are injA\B\Cj:3. An insurance company believes that people can be divided into two classes: thosewho are accident prone and those who are not. Assume that 30% of the population isaccident prone. The company’s statistics show that an accident-prone person will havean accident at some time within a ±xed 1-year period with probability 0:4, whereas thisprobability is only 0:2 for a non-accident-prone person.(a) What is the probability that a new policyholder will have an accident within ayear of purchasing a policy?(b) Given that a new policy holder has an accident within a year of purchasing apolicy, what is the conditional probability that he or she is accident-prone? (2 points)4. IfP(E) = 0:9 andP(F) = 0:8, show thatP(E\F)°0:7.5. It is known from experience that in a certain industry 60% of all labor-managementdisputes are over wages, 15% are over working conditions, and 25% are over fringe issues.Also, 45% of the disputes over wages are resolved without strikes, 70% of the disputesover working conditions are resolved without strikes, and 40% of the disputes over fringeissues are resolved without strikes.What is the probability that a labor-managementdispute in industry will be resolved without a strike? Knowing that a labor-managementdispute in industry is resolved without a strike, what is the probability that the disputewas over working conditions?6. Suppose that there are 14 songs on a compact disk (CD) and you like 8 of them.When using the random button selector on a CD player, each of the 14 songs is playedonce in a random order.Find the probability that among the ±rst two songs that areplayed,(a) You like both of them. (b) You like exactly one of them.7.Suppose a random variableXhas Beta distribution with a probability density func-tion given byf(x) =(cx2(1±x)for 0²x²10elsewherea) Find the value ofcthat makes a probability density function (PDF).b) FindPr(0:3:²X²1); c) FindPr(X²0:3jX²0:8); d) FindF(b) = Pr(X²b)and sketch the graph of this function.