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UCLA STATS 100A - classhandout-key

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1 2. Conditional Probability Exercise 2 Days on the market Selling price Less than 30 days 30 – 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - $600,000 35 45 4 84 Over $600,000 8 4 0 12 Total 82 80 19 181 1. For the 181 house sales in the above table: (a) What proportion of the houses that sold for under $300,000 were on the market for less than 30 days? ______________________________________________________________________ (b) Given that a house sold for between $300,000 and $600,000 (inclusive), what proportion were on the market for more than 90 days? ______________________________________________________________________ (c) If a house was on the market for less than 30 days, what proportion sold for over $600,000? ______________________________________________________________________ (d) Of those on the market for more than 90 days, what proportion sold for under $300,000? ______________________________________________________________________2 2. For the 181 house sales in the above table: (a) What proportion of the houses took more than 90 days to sell? ______________________________________________________________________ (b) What proportion sold for under $300,000 given that they were on the market for 30 to 90 days (inclusive)? ______________________________________________________________________ (c) What proportion took less than 30 days to sell or sold for between $300,000 and $600,000 (inclusive)? ______________________________________________________________________ (d) Of those that sold for over $600,000, what proportion were on the market for more than 90 days? ______________________________________________________________________ (e) What proportion sold for between $300,000 and $600,000 (inclusive) and took from 30 to 90 days (inclusive) to sell? ______________________________________________________________________ (f) What proportion of the houses took 90 days or less to sell? ______________________________________________________________________ (g) What proportion of the houses that were on the market for more than 90 days sold for over $600,000? ______________________________________________________________________ (h) What proportion sold for over $600,000 and took more than 90 days to sell? ______________________________________________________________________ 3. Comment on your answers to Questions 2(d) and 2(g). ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ E and H are mutually exclusive events. Both conditional probabilities are zero. This is because if E has happened then H cannot occur and vice versa.3 4. (a) Fill in the gaps: Probability a house sold for under $300,000 given that it sold in less than 30 days = = = = (b) Fill in the gaps: Probability a house sold for between $300,000 and $600,000 (inclusive) given that it was on the market for between 30 and 90 days (inclusive) = = = = (c) Fill in the gaps: For any two events A and B, P(A | B) = C F 39 82 F D G 45 80 G B 39 C and F 82 45 804 4. Independent Events If A and B are independent events then P(A | B) = ______________ Exercise 4 Days on the market Selling price Less than 30 days 30 – 90 days More than 90 days Total Under $300,000 39 31 15 85 $300,000 - $600,000 35 45 4 84 Over $600,000 8 4 0 12 Total 82 80 19 181 1. Recall: C is the event that a house sold for under $300,000 and F is the event that the sale was made in less than 30 days. Are events C and F independent? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2. Recall: E is the event that a house sold for over $600,000 and H is the event that the sale was made in more than 90 days. Are events E and H independent? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Discussion Exercise Fill the gaps: For events A and B, = If A and B are independent events, = P( ) So if A and B are independent events, = P( ) That is, if A and B are independent events P( ) = P( ) x P( ) P(A) C and F are not independent because P(C | F) ≠ P(C) E and H are not independent because P(E | H) ≠ P(E) B A B A A B5 Exercise 5 1. For events A and B, P(A) = 0.8, P(B) = 0.5 and P(A or B) = 0.9. (a) Calculate P(A and B) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ (b) Calculate P(A | B) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ (c) Are A and B independent events? Justify your answer. ________________________________________________________________________ 2. For events A and B, P(A) = 0.3, P(B) = 0.4 and P(A | B) = 0. (a) Are A and B independent events? Justify your answer. ________________________________________________________________________ (b) Are A and B mutually exclusive events? Justify your answer. ________________________________________________________________________ 3. Events A and B are independent. Also P(A | B) = 0.4 and P(B) = 0.6. (a) State P(A) ________________________________________________________________________ (b) Calculate P(A and B) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ = 0.8 Yes, because P(A | B) = P(A) No, because P(A | B) ≠ P(A) Yes, because P(A | B) = 0 P(A) = 0.4 (by independence, P(A | B) = P(A)) = 0.24 P(A or B) = P(A) + P(B) – P(A and B)


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UCLA STATS 100A - classhandout-key

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