Practice Problems for Exam 1 1 a The local newsagent has gift wrapping paper and ribbon on sale The paper comes in green blue purple and white and the ribbon comes in red and gold If you are buying one package of paper and one of ribbon how many different color combinations can you choose b How many different signals can be sent by displaying sequences of three flags on a mast if there are six different flags available c How many distinct sets of 3 differently colored scarves can be bought if the shop has scarves in 8 different colors d Among fourteen orchids for a line of orchids along one wall three are white four lavender two yellow one pink and four orange What is the number of different color displays 2 In answering questions on a multiple choice test a student either knows the answer or guesses Let p be the probability that he knows the answer and 1 p that he guesses Assume that a student who guesses at the answer will be correct with probability 1 m where m is the number of multiple choice alternatives A student who knows the answer will answer the question correctly What is the conditional probability that the student knew the answer to a question given that he answered it correctly 3 The cumulative distribution function of a random variable X is given by x 2 3 0 1 2 2 3 x 2 2 3 2 x 4 7 5 6 4 7 x 6 1 x 6 F x a Find and tabulate the pmf of X b Compute P 2 X 5 and P 2 X 5 c Find the mean and variance of X d Find the mean and variance of 5 3X e Find the expectation of eX 4 A batch consists of 15 coil springs from a production process that are checked for conformance to customer requirements The mean number of nonconforming coil springs in a batch is 3 Assume that conformance statuses of different springs are independent a Find the probability that the number of nonconforming springs in a batch is greater than 1 but less than 4 b Find the standard deviation of the number of nonconforming springs in a batch 5 The probability of a successful optical alignment in a trial is 0 6 Assume that the trials are independent a What is the probability that the first successful alignment requires at most b What is the probability that it requires 10 trials to get the third successful four trials alignment 6 Arrivals of telephone calls is often modeled as a Poisson process Suppose on the average there are 5 calls per hour a Find the probability that there are more than 3 calls in 30 minutes b If no call has come in the last 30 minutes find the probability that at least one call comes in the next 15 minutes 7 Suppose that 1 of the residents of New York City believe that the incident in Roswell New Mexico in July of 1947 was related to UFO In a survey 500 NYC residents were asked if they believe that the Roswell incident was related to UFO Approximate the probability that not more than 7 people answered that the Roswell incident was related to UFO State and check conditions for the approximation 8 Let E F G be three independent events with probabilities P E 0 1 P F 0 3 P G 0 5 For each of the following events find the expression in terms of E F G and union intersection complement Venn diagram and the probability 9 A maintenance firm has gathered the following information regarding the failure a only F occurs b at most two events occur mechanisms for air conditioning systems Electrical failure Gas leak yes no 17 3 yes no 55 32 a Find the probability there is an electrical failure given that there was a gas leak b If there is an electrical failure find the probability that there is a gas leak