University of California, Los AngelesDepartment of StatisticsStatistics 110A Instructor: Nicolas ChristouHomework 3EXERCISE 1An unbiased coin is flipped 5 times. The probability of observing heads on any toss is12.a. Construct the probability distribution of X, where X is the number of heads observed in these 5 tosses.b. What is the probability of observing exactly 2 heads given that in these 5 tosses at least 1 head occurred?EXERCISE 2A committee of five members is to be formed from a group of 6 men and 4 women.a. Construct the probability distribution of X, where X is the number of women in the committee.b. Find the probability that the majority of the committee will be women.EXERCISE 3In a game of darts, the probability that a particular player aims and hits treble twenty with one dart is 0.40. How many throwsare necessary so that the probability of hitting the treble twenty at least once exceeds 90%?EXERCISE 4In a particular department store customers arrive at a checkout counter according to a Poisson distribution at an average ofseven per hour. During a given hour, what are the probabilitiesa. that no more than three customers arrive?b. that at least two customers arrive?c. that exactly five customers arrive?EXERCISE 5New York Lotto is played as follows: Out of 59 numbers 6 are chosen at random without replacement. Then from the remaining53 numbers 1 is chosen. This last number is called “the bonus number”. You, the player, select 6 numbers. To win the firstprize you must match your 6 numbers with the State’s 6 numbers. If you match only 5 numbers and your 6thnumber matchesthe bonus number then you win the second prize.a. What is the probability of winning the first prize?b. What is the probability of winning the second prize?c. What is the probability of winning a prize (either the first or the second)?Note: Check your answers to (a, b) at http://www.nylottery.org/games/lotto.php .EXERCISE 6A particular industrial product is shipped in lots of 20. Testing to determine whether an item is defective is costly, and hence themanufacturer samples the production. A sampling plan constructed to minimize the number of defectives shipped to customerscalls for sampling and testing 5 items from each lot and rejecting the lot if more than 1 defective is observed. Suppose that alot contains 4 defectives.a. Compute the probability that the test will find 0 defectives.b. Compute the probability that the test will find 1 defective.c. Compute the probability that the lot will be rejected.EXERCISE 7A missile protection system consists of n radar sets operating independently, each with a probability of 90% of detecting amissile entering a zone that is covered by all of the units.a. Suppose that a protection system consists of n = 5 radars sets. Construct the probability distribution of the numberof radar sets that detect a missile that enters the zone.b. Compute the expected value and the standard deviation of the number of radar sets that detects a missile that entersthe zone (n = 5).c. How large must n be if we require that the probability of detecting a missile that enters the zone be 99.9%?EXERCISE 8Let X be a geometric random variable with probability of success p. The probability mass function of X is:P (X = k) = (1 − p)k−1p, k = 1, 2, · · ·Show that the probabilities sum up to 1.EXERCISE 9A satellite system consists of n components and functions on any given day if at least k of the n components function onthat day. On a rainy day each of the components independently functions with probability p1, whereas on a dry day theyindependently function with probability p2. The probability of a rainy day is θ.a. Give an expression of the probability that the satellite system will function at any given day.b. Evaluate the above probability when n = 3, k = 1, p1= 0.85, p2= 0.90, and θ = 0.20.c. If the satellite system functioned yesterday what is the probabilty that it was a rainy day? Use n = 3, k = 1, p1= 0.85,p2= 0.90, and θ = 0.20.EXERCISE 10Let X be a geometric random variable with probability of success p.a. Shaw that for a positive integer k, P (X > k) = (1 − p)k).b. Show that for positive integers a and b,P (X > a + b|X > a) = (1 − p)b= P (X >