W2-slides3 - Lecture 7- Outline and Examples ( 3.4 in Ross)...

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Unformatted text preview: Lecture 7- Outline and Examples ( 3.4 in Ross) Independence- Two independent events-More than two independent events-Pairwise independent events Random variables Independent Events (Ross 3.4) Recall Multiplication rule: P ( A B ) = P ( A | B ) P ( B ) = P ( B | A ) P ( A ) . Independent Events (Ross 3.4) Definition. Two events A and B are said to be independent if P ( A B ) = P ( A ) P ( B ) or equivalently, P ( A | B ) = P ( A ) or P ( B | A ) = A . Independent Events (Ross 3.4) Definition. Two events A and B are said to be independent if P ( A B ) = P ( A ) P ( B ) or equivalently, P ( A | B ) = P ( A ) or P ( B | A ) = A . Intuitively , independent events have no influence on each other. For example, getting a head in the first toss of a coin does not influence an outcome of the second toss of the coin. Example-independence Example. Two draws from a well-shuffled deck of cards. Event A: get an ace in the first draw; Event B: get an ace in the second draw. Problem 1 (draws done with replacement) . We take the first card, look at it and put it back into the deck. Deck is reshuffled, then the second card is drawn. Are events A and B independent? Because this problem has two stages in it, we use conditional probability based definition. draws with replacement- independent Answer. Yes. Compute P ( B | A ). After the first draw, we replace the card. Even if we draw an ace the first time, we will replace it, and so at the second draw we will still have 52 cards and 4 aces. P ( B | A ) = P(an ace out of 52 cards; 4 aces in it) = 4/52 = P(B). draws without replacement-dependent Problem 2 (draws done without replacement) . We take the first card, then the second. Are events A and B independent?...
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W2-slides3 - Lecture 7- Outline and Examples ( 3.4 in Ross)...

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