EXAMPLE: HOMEWORK 3: PROBABILITIES Justify (explain) your answers Due: Sunday, September 13 For full credit, show all your work, justify your answers Before starting to solve this homework, review...

Free Quote


EXAMPLE: HOMEWORK 3: PROBABILITIES Justify (explain) your answers Due: Sunday, September 13 For full credit, show all your work, justify your answers Before starting to solve this homework, review your lecture notes and pay attention to problem 1’s solution. Justify your answers. Show the literal formula and not just the final result. PROBLEM A survey examines customers’ preferences in having a sunroof in their car. Sunroof (S) No Sunroof (S’) Woman (W) 800 650 1450 Man (M) 1500 300 1800 2300 950 3250 a) Compute the Marginal Probabilities and the Joint Probabilities. b) Compute: P(W), P(S|M), P(S’|W), P(M and S), P(W or S). c) Is there any relationship between sex (woman or man) and preferring to have a sunroof? SOLUTION a) Compute the Marginal Probabilities and the Joint Probabilities. Marginal Joint P(S) = 2300/3250 = 71% P(S and W) = 800/3250 = 25% P(S’) = 950/3250 = 29% P(S and M) = 1500/3250 = 46% P(W) = 1450/3250 = 45% P(S’ and W) = 650/3250 = 20% P(M) = 1800/3250= 55% P(S’ and M) = 300/3250 = 9% b) Compute: P(W), P(S|M), P(S’|W), P(M and S), P(W or S) . P(W) =45% P(S|M) = P(S and M)/P(M) = 46%/55% = 84% P(S’|W) = P(S’ and W)|P(W)=44% P(M and S) =46% P(W or S) = P(W)+P(S)-P(W and S) = 45% + 71% - 25% = 91% c) Is there any relationship between sex (woman or man) and wanting to have a sunroof? P(S) = ? P(S | W) =? P(S|M) P(S) = 71%P(S|W) = 55% P(S|M) = 84% If we don’t have equality among the above probabilities, then there IS a relationship. If all the above probabilities would have been equal, then it would have been no relationship. In Conclusion: There is a greater probability of owning a car with sunroof given you are a man. Note: discussing relationship should be done based on the given probabilities values (A|B) and not just yes or no. Another way of testing: Multiplication rule. If two events ARE INDEPENDENT then: P(A and B) = P(A)*P(B) // P(A and B) = P(A|B) P(B) = P(B|A) P(A) Note: If you have an intersection between the events do not use the above rule. Addition Rule. The correct way to compute P(A or B): P(A or B) = P(A) + P(B) – P(A and B) PROBLEM 1 (25 points): Dishwasher and singles S (single) S′ (not single) D (own dishwasher) 10 500 D’ (no dishwasher) 75 200 a. Compute the Marginal Probabilities and the Joint Probabilities. b. What is the probability of owning a dishwasher? c. What is the probability of being single knowing that you own a dishwasher? d. What is the probability of not owning a dishwasher given that one is not single? e. Are dishwasher ownership and being single events related? PROBLEM 2 (20 points): The Gift Basket Store had the following premade gift baskets containing the following combinations in stock: Cookies Mugs Candy Coffee 20 13 10 Tea 12 10 12 Choose one basket at random. Find the probability that it contains a) Tea or Cookies b) Coffee given that it contains Candies c) Tea and Mugs d) Is there any relation between picking cookies and the type of drink (coffee, tea)? (consider P(cookies|coffee), P(cookies|tea), P(cookies)) PROBLEM 3: (15 points) Educational level and Smoking At a large factory the employees were surveyed and classified according to their level of education and whether they smoked. Not HS graduate HS graduate College Graduate Smoke 21 11 14 Do not smoke 3 9 20 If an employee is selected at random, find these probabilities: a) The employee smokes, given that he or she graduated from college. b) Given that the employee did not graduate from high school, he or she is a smoker. c) Is there any relationship between education and the smoking habit? Explain it based on the probabilities’ values. PROBLEM 4 (30 POINTS) – given in a previous midterm Some collected data is presented in the table below: Age Category Team {18-20} {21 – 25} {26-30} Above 30 JumpingHigh (JH) 10 15 11 8 SpeedySkies (SS) 3 5 12 30 WinterFun (WF) 15 10 10 6 Give the literal formula first (not with numbers) and then solve: “What is the probability of being in the highest age category given that you are a SpeedySkies member” Give the literal formula first (not with numbers) and then solve: “What is the probability of being a SpeedySkies member ” in the {21-25} range of age Give the literal formula first (not with numbers) and then solve: “What is the probability of being between 18 and 25 years old given that you are a member of JumpingHigh team”. Give the literal formula first (not with numbers) and then solve: “What is the probability of not being a member of WinterFun team” Is there any relationship between being a member in the {18-20} age category and belonging to a specific team? (relationship between age and team)