8. and 9.2% are both blue and have a collar. A random, shirt from the factory is

Question

8. and 9.2% are both blue and have a collar. A random, shirt from the factory is selected. A factory manufactures m any shirts, of which 24.2% are blue, 67.1% have a collar a. Find the probability that the shirt is blue or has a collar Find the probability that the shirt is blue given that it has a collar. c. Find the probability that the shirt is blue and does.not have a collar. d. Find the probability that the shirt has a collar given that it is NOT blue. e. Are being blue and having a collar independent events? Justify 9. In a population of 20-29 year-old males, 80% do not smoke and 20% do smoke Of the non-smokers 15% have asthma and of the smokers 77% have asthma. A randomly selected 20-29 year old male from this population is selected. a. Find the probability that he does not have asthma GIVEN he does not smoke. b. Find the probability that he smokes AND has asthma. c. Find the probability that he has asthma. d. Find the probability that he smokes GIVEN he has asthma. e. Find the probability that he does not smoke GIVEN he has asthma. 10. There are three runners, Abe, Bob and Carl. They each run the 800 m race. Abe is 10 years-old and the average time for his age is 4.11 minutes with a standard deviation of 1.25 minutes. Bob is 11 years-old and the average time for his age is 3.75 minutes with a standard deviation of 1.12 minutes. Carl is 12 years-old and the average time for his age is 3.33 minutes with a standard deviation of 1.07 minutes. Abe runs the 800 m in 4.00 minutes, Bob runs the 800 m in 3.50 minutes and Carl runs the 800 m in 2.87 minutes. Who is the fastest runner with respect to their age group? Justify! 11. The working lifetime of a certain type of circuit has a Uniform distribution with minimum of 50 and maximum 150 days. a. Find the mean and standard deviation of the lifetimes b. Find the probability that a single circuit lasts between 60 and 75 days c. Find the 80th percentile of the circuit lifetimes. d. Given that the circuit has lasted at least 70 days, find the probability that it lasts less than 100 days.

Explanation / Answer

8) P(Blue) = 0.242, P(Collar) = 0.671 and P(Blue and Collar) = 0.092

a) P(the shirt is blue or has a collar) = P(blue) + P(collar) - P(Blue and collar)
= 0.242 + 0.671 - 0.092 = 0.821
b) P(shirt is blue / it has a collar) = P(Blue and collar) / P(collar)
= 0.092 / 0.671 = 0.1371

c) P(blue and not collar ) = P(Blue) - P(blue and not collar ) = 0.242 - 0.092 = 0.15

d) P(collar and not blue ) = P(collar) - P(collar and blue ) = 0.671 - 0.092 = 0.579
P(shirt has collar / it is not blue) = P(collar and not blue )/ P(not blue) = 0.579 / (1-0.242) = 0.7639

e) P(Blue) * P(Collar) = = 0.242 * 0.671 = 0.1624 not equal to P(Blue and Collar) = 0.092
Therefore Blue and Collar are not independent

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