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**Chapter 14: From Randomness to Probability**

by Royce Hong and Kenneth Wang

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Vocabulary Random Phenomenon: a situation in which we know what outcomes could happen, but the particular outcome is uncertain Probability: the long run relative frequency of an event Trial: each attempt of a random phenomenon Outcome: the result or value of a trial Event: a combination of outcomes

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Vocabulary Continued Independent: when the outcome of one trial does not influence the outcome of another Dependent: when the outcome of one trial does affect the outcome of another Law of Large Numbers: states that the long-run relative frequency of repeated independent events approaches the true relative frequency as the number of trials increases.

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Vocabulary Continued Personal Probability: also referred to as subjective probability, probability expressed with a degree of uncertainty in everyday speech without basing it on long-run frequencies, not consistent like formal probabilities

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Formal Probability For any event A, 0 ≤ P(A) ≤ 1 or the probability of event A occurring lies between 0 and 1. P(S) = 1 or the probability of all sets of possible outcomes together is 1. The set of outcomes that are not within event A are called the complement of A, or Ac. P(A) = 1 – P(Ac) or the probability of an event occurring is 1 minus the probability that it doesn’t occur.

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**Formal Probability Continued**

Events that have no outcome in common are called disjoint or mutually exclusive. P(A or B) = P(A) + P(B) or the probability that one or the other occurs is the sum of the probabilities of the two events.

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**Formal Probability Continued**

For two independent events A and B, P(A and B) = P(A) x P(B) or the probability that both A and B occur is the product of the probabilities of the two events. P(A U B) is the same as P(A or B) . P(A ∩ B) is the same as P(A and B).

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**What can go wrong? Beware of probabilities that don’t add up to one.**

Don’t add probabilities of events if they’re not disjoint. Don’t multiply probabilities of events if they’re not independent. Don’t confuse disjoint and independent.

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Problem 29 A certain bowler can bowl a strike 70% of the time. What is the probability that she goes three consecutive frames without a strike? makes her first strike in the third frame? has at least one strike in the first three frames? bowls a perfect game (12 consecutive strikes)?

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Solution to Problem 29a The outcome of a frame is independent of the outcomes of other frames. Therefore, using the Multiplication Rule and Complement Rule, P(3 consecutive without strike)= P(without strike)3 = (1 - P(strike))3 = (0.30)3 = 0.027

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Solution to Problem 29b The outcome of a frame is independent of the outcomes of other frames. Getting the first strike in the third frame means not getting a strike in the first two frames. Using the Complement Rule and the Multiplication Rule, P(first strike in third)= P(no strike)*P(no strike)*P(strike) = ( )( )(0.70) = 0.063

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Solution to Problem 29c The outcome of a frame is independent of the outcomes of other frames. Using the Complement Rule, the probability of at least one strike in the first three frames is complement to the probability of no strikes in the first three frames. Therefore, using the Multiplication Rule, P(at least one in three)= 1 - P(no strikes in three) = 1 - ( )*( )*( ) = 1 - (0.30)3 = 0.973

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Solution to Problem 29d The outcome of a frame is independent of the outcomes of other frames. Using the Multiplication Rule, P(12 consecutive strikes) = P(strike)12 = (0.70)12 = 0.014

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Problem 31 Suppose that in your city 37% of the voters are registered as Democrats, 29% as Republicans, and 11% as members of other parties (Liberal, Right to Life, Green,etc.). Voters not aligned with any official party are termed “Independent.” You are conducting a poll by calling registered voters at random. In your first three calls, what is the probability you talk to all Republicans? no Democrats? at least one Independent?

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Solution to Problem 31a The outcome of the phone call is independent of the outcomes of other phones calls. Using the Multiplication Rule, P(3 Republicans) = P(Republican)3 = (0.29)3 = 0.024

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Solution to Problem 31b The outcome of the phone call is independent of the outcomes of other phones calls. Using the Multiplication Rule and the Complement Rule, P(no Democrats) = (1 - P(Democrat))3 = ( )3 = (0.63)3 = 0.250

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Solution to Problem 31c The outcome of the phone call is independent of the outcomes of other phones calls. The probability of at least one Independent is complement to the probability of no Independent. Using the Multiplication and Complement Rule, P(at least one Independent) = 1 - P(no Independent) = 1 - (1 - P(Independent))3 = 1 - ( )3 = 0.543

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