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Independent Events

Independent Events. Two outcomes are independent if knowing the outcome of one does not change the probabilities for outcomes of the other. When two events are independent, we find the probability of both events happening by multiply their individual probabilities.

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Independent Events

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  1. Independent Events • Two outcomes are independent if knowing the outcome of one does not change the probabilities for outcomes of the other. • When two events are independent, we find the probability of both events happening by multiply their individual probabilities.

  2. Example with Independent Events • Suppose you are taking a multiple choice test where each question has 4 choices. • There are two problems that you do not know the answers to. • You decide to guess on these two questions. • What are your chances of answering both questions correctly?P(correct) = ¼ = .25. Each question is independent.

  3. Chapter 20: The House Edge: Expected Values • Gamblers want to know what they can expect to win (or lose) in a game of chance. • Example: The old New York Lottery. For each one-million tickets sold: 1 $50,000 prize 9 $5,000 prizes 90 $500 prizes 900 $50 prizes 999,000 $0 prizes (losing tickets)

  4. Expected Value • The expected value of a random phenomenon that has numerical outcomes is found by multiplying each outcome by its probability and then summing over all possible outcomes. (p. 394) • In symbols, the possible outcomes are a1, a2, …, ak that have the respective probabilities of p1, p2, …, pk, the expected value is: Expected Value = a1p1 + a2p2 + … + akpk • Can interpret expected value as the average value in the long run; over many trials.

  5. Lottery Ticket Example • Find the expected value (payout) of a ticket. AmountProbabilityDecimalaP(a)$50,000 1/1,000,000 = 0.000001 0.05$5,000 9/1,000,000 = 0.000009 0.045$500 90/1,000,000 = 0.00009 0.045$50 900/1,000,000 = 0.0009 0.045$0 999,000/1,000,000 = 0.999 0 • When New York had this lottery, a ticket cost 50¢. • What are the expected earnings of a ticket?

  6. Expected Value of a Die Roll • What is the expected value of a roll of a die? • Recall: Each number has probability = 1/6 • Possible numbers: 1, 2, 3, 4, 5, 6 Expected Value = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5 • Note: In both examples, we see that the expected value is not a possible outcome.

  7. Law of Large Numbers • If a random phenomenon with numerical outcomes is repeated many times independently, the mean value of the actually observed outcomes approaches the expected value. (p. 396) • For Theoretical Probabilities, the expected value can be calculated exactly. • For Experimental Probabilities, we can find approximate expected values by simulation or experimentation.

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