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(i) Show that the probability that there will be no collision in a five-day week is (1 – p)°. (ii) State one assumption that is made in your answer above. (iii) Let p = 0.001. Check that the Poisson approximation can be used, and find the approximate probability that Alan will avoid a collision in 500 working days. Round

(i) Show that the probability that there will be no collision in a five-day week is (1 – p)°. (ii) State one assumption that is made in your answer above. (iii) Let p = 0.001. Check that the Poisson approximation can be used, and find the approximate probability that Alan will avoid a collision in 500 working days. Round

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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(a) Every working day, Alan reverses his car from his driveway on to the road in such a way that there is a very small probability p that his car will be involved in a collision. (i) Show that the probability that there will be no collision in a five-day week is (1– p)³. (ii) State one assumption that is made in your answer above. (iii) Let p 0.001. Check that the Poisson approximation can be used, and find the approximate probability that Alan will avoid a collision in 500 working days. Round your answer to 4 decimal places. (b) Organisms are present in ballast water discharged from a ship according to a Poisson pro- cess with a concentration of 4 organisms per cubic meter. Find the probability that the number of organisms in 1.5 cubic meter of discharge exceeds its mean value by more than one standard deviation. Round your answer to 4 decimal places. (c) The number of times that a person gets a cold in a given year is a Poisson random variable with parameter A = 5. Suppose that a new supplement which has just been marketed reduces the Poisson parameter to A = 3 for 75% of the population. For the other 25% of the population, the supplement gives no significant reduction on the number of colds. If an individual tries the supplement for a year and has 2 colds in that time, find the conditional probability that the supplement is beneficial for him or her. Round your answer to 4 decimal places.
Transcribed Image Text:(a) Every working day, Alan reverses his car from his driveway on to the road in such a way that there is a very small probability p that his car will be involved in a collision. (i) Show that the probability that there will be no collision in a five-day week is (1– p)³. (ii) State one assumption that is made in your answer above. (iii) Let p 0.001. Check that the Poisson approximation can be used, and find the approximate probability that Alan will avoid a collision in 500 working days. Round your answer to 4 decimal places. (b) Organisms are present in ballast water discharged from a ship according to a Poisson pro- cess with a concentration of 4 organisms per cubic meter. Find the probability that the number of organisms in 1.5 cubic meter of discharge exceeds its mean value by more than one standard deviation. Round your answer to 4 decimal places. (c) The number of times that a person gets a cold in a given year is a Poisson random variable with parameter A = 5. Suppose that a new supplement which has just been marketed reduces the Poisson parameter to A = 3 for 75% of the population. For the other 25% of the population, the supplement gives no significant reduction on the number of colds. If an individual tries the supplement for a year and has 2 colds in that time, find the conditional probability that the supplement is beneficial for him or her. Round your answer to 4 decimal places.
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