Chapter 5 - Practice Problems 3

Transcription

1 Chapter 5 - Practice Problems 3 Evaluate the expression. 12 1) 2 1) A) 3,628,800 B) 66 C) 7,257,600 D) 6 2) 20 1 A) 20 B) 21 C) 1 D) 19 2) Solve the problem. 3) A coin is biased so that the probability it will come up tails is The coin is tossed three times. Considering a success to be tails, formulate the process of observing the outcome of the three tosses as a sequence of three Bernoulli trials. Complete the table below by showing each possible outcome together with its probability. Display the probabilities to three decimal places. List the outcomes in which exactly two of the three tosses are tails. Without using the binomial probability formula, find the probability that exactly two of the three tosses are tails. 3) hhh (0.53)(0.53)(0.53) = ) 30% of the adult residents of a certain city own their own home. Four residents are selected at random from the city and asked whether or not they own their own home. Considering a success to be ʺowns their own homeʺ, formulate the process of observing whether each of the four residents owns their own home as a sequence of four Bernoulli trials. Complete the table below by showing each possible outcome together with its probability. Display the probabilities to three decimal places. List the outcomes in which exactly two of the four residents own their own home. Without using the binomial probability formula, find the probability that exactly two of the four residents own their own home. 4) ssss (0.3)(0.3)(0.3)(0.3) = Find the indicated binomial probability. 5) What is the probability that 6 rolls of a fair die will show four exactly 2 times? 5) A) B) C) D) ) A company manufactures calculators in batches of 64 and there is a 4% rate of defects. Find the probability of getting exactly 3 defects in a batch. 6) A) 2.88 B) C) D)

2 Find the indicated probability. 7) A test consists of 10 true/false questions. To pass the test a student must answer at least 7 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test? 7) A) B) C) D) ) A machine has 7 identical components which function independently. The probability that a component will fail is 0.2. The machine will stop working if more than three components fail. Find the probability that the machine will be working. 8) A) B) C) D) ) An airline estimates that 96% of people booked on their flights actually show up. If the airline books 80 people on a flight for which the maximum number is 78, what is the probability that the number of people who show up will exceed the capacity of the plane? 9) A) B) C) D) Find the mean of the binomial random variable. 10) The probability that a person has immunity to a particular disease is 0.6. Find the mean for the random variable X, the number who have immunity in samples of size ) A) 15.6 B) 0.6 C) 10.4 D) ) In a certain town, 90 percent of voters are in favor of a given ballot measure and 10 percent are opposed. For groups of 220 voters, find the mean for the random variable X, the number who oppose the measure. 11) A) 22 B) 90 C) 198 D) 10 Find the specified probability distribution of the binomial random variable. 12) In one city, 21% of the population is under 25 years of age. Three people are selected at random from the city. Find the probability distribution of X, the number among the three that are under 25 years of age. A) B) ) C) D)

3 Construct a probability histogram for the binomial random variable, X. 13) A baseball player batting comes to bat 4 times in a game. X is the number of hits. A) B) 13) C) D) Provide an appropriate response. 14) List the four requirements for a binomial distribution. Describe an experiment which is binomial and discuss how the experiment fits each of the four requirements. 14) 15) Three random variables X, Y, and Z, are described below. In which of these situations would it be acceptable to use the binomial distribution? A: A bag contains 4 blue marbles and 8 red marbles. Five marbles are drawn at random with replacement. The random variable X is the number of blue marbles drawn. B: A bag contains 4 blue marbles and 8 red marbles. Six marbles are drawn at random without replacement. The random variable Y is the number of blue marbles drawn. C: A bag contains 30 blue marbles and 38 red marbles. Three marbles are drawn at random without replacement. The random variable Z is the number of blue marbles drawn. A) A and C B) A only C) A, B, and C D) B and C 15) 16) Identify each of the variables in the Binomial Formula. n! P(x) = (n - x)!x! p x (1-p)n-x Also, explain what the fraction n! (n - x)!x! computes. 16) 3

4 Answer Key Testname: CH 5 SET 3 1) B 2) A 3) Each trial consists of observing whether the coin comes up heads or tails. There are two possible outcomes, heads or tails. The trials are independent. If we consider tails to be success, the success probability is p = hhh (0.53)(0.53)(0.53) = hht (0.53)(0.53)(0.47) = hth (0.53)(0.47)(0.53) = htt (0.53)(0.47)(0.47) = thh (0.47)(0.53)(0.53) = tht (0.47)(0.53)(0.47) = tth (0.47)(0.47)(0.53) = ttt (0.47)(0.47)(0.47) = htt, tht, tth; ) Each trial consists of observing whether or not the resident owns their own home. There are two possible outcomes, owns their own home or does not own their own home. The trials are independent. If we consider ʺowns their own homeʺ to be success, the success probability is p = 0.3. ssss (0.3)(0.3)(0.3)(0.3) = sssf (0.3)(0.3)(0.3)(0.7) = ssfs (0.3)(0.3)(0.7)(0.3) = ssff (0.3)(0.3)(0.7)(0.7) = sfss (0.3)(0.7)(0.3)(0.3) = sfsf (0.3)(0.7)(0.3)(0.7) = sffs (0.3)(0.7)(0.7)(0.3) = sfff (0.3)(0.7)(0.7)(0.7) = fsss (0.7)(0.3)(0.3)(0.3) = fssf (0.7)(0.3)(0.3)(0.7) = fsfs (0.7)(0.3)(0.7)(0.3) = fsff (0.7)(0.3)(0.7)(0.7) = ffss (0.7)(0.7)(0.3)(0.3) = ffsf (0.7)(0.7)(0.3)(0.7) = fffs (0.7)(0.7)(0.7)(0.3) = ffff (0.7)(0.7)(0.7)(0.7) = ssff, sfsf, sffs, fssf, fsfs, ffss; ) B 6) B 7) A 8) B 9) D 10) A 11) A 12) A 13) C 4

5 Answer Key Testname: CH 5 SET 3 14) The four requirements are: 1) The experiment must have a fixed number of trials. 2) The trials must be independent 3) Each trial must have all outcomes classified into two categories. 4) The probabilities must remain constant for each trial. Answers will vary for the experiment. 15) B 16) n is the fixed number of trials, x is the number of successes, p is the probability of success in one of the n trials, and (1 - p) is the probability of failure in one of the n trials. The fraction determines the number of different orders of x successes out of n trials. 5