03_02 - Example (Gold Coins): The Story. Box 1 contains 2...

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p. 3-11 Example (Gold Coins): The Story. Box 1 contains 2 silver coins. Box 2 contains 1 gold and 1 silver coin. Box 3 contains 2 gold coins. Experiment: (i) Select a box at random and, (ii) Examine the 2 coins in order (assuming all choices are equally likely at each stage) Q : Given that 1 st coin is gold, what is the probability that Box k is selected, k =1, 2, 3? Let A k = {Box k is selected}, B = {1 st coin is gold}, P ( B | A k )= 0 , if k =1 , 1 / 2 , if k =2 , 1 , if k =3 . P ( B 1 3 · 0+ 1 3 · 1 2 + 1 3 · 1= 1 2 . P ( A 1 | B (1 / 3) · 0 1 / 2 =0 . Similarly, P ( A 2 | B )=1 / 3 , P ( A 3 | B )=2 / 3 . p. 3-12 P ( B C | A k 0 , if k , 0 , k , 1 , k . P ( B C 1 3 · 1 3 · 1 3 · 1 3 . Q : Given that 1 st coin is gold, what is the probability that 2 nd coin is gold? Let C = {2 coin is gold}. P ( C | B P ( B C ) P ( B ) = 1 / 3 1 / 2 = 2 3 . Q : Why are the 3 formulas useful in calculating probabilities? (Note: They all benefit from conditional probabilities.) Ans : (i) & & & ; (ii) =conditioning because the sample space is reduced from to a smaller set. (For example, in many cases, P ( B | A j )’s are known or are easier to find) The odds of an event A : • Odds and Conditional Odds
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p. 3-13 Reading : textbook, Sec 3.1, 3.2, 3.3, 3.5 The odd of event B given A : and Independence • Definition (independence for 2-events case): Two events A and B are said to be independent if and only if Otherwise, they are said to be dependent .
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03_02 - Example (Gold Coins): The Story. Box 1 contains 2...

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