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Chapter 17:Thinking about Chance

Chapter 17:Thinking about Chance. Randomness of probability (p. 348) A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.

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Chapter 17:Thinking about Chance

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  1. Chapter 17:Thinking about Chance Randomness of probability (p. 348) • A phenomenon is random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions. • The probability of any outcome of a random phenomenon is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions. • An impossible outcome has probability of 0 and a certain outcome has probability of 1.

  2. Two types of probabilities: • A event refers to a collection of outcomes. • Experimental Probability: Look at outcomes of an experiment repeated a large number of times to determine proportion of occurrences of particular events. • Theoretical Probability: For equally likely events,P(event) = # of ways the event can occur # of possible outcomes

  3. Examples of random phenomena that lead to probabilities: Flipping a coin: Possible outcomes are heads or tails. P(H) = ½ and P(T) = ½. Theory or Experiment? Tossing a die: Possible numbers are 1, 2, 3, 4, 5, or 6 P(any one number)=1/6. P(even number)=3/6 =1/2 Theory or Experiment? Having a baby: Possible outcomes are boy or girl. Past data shows P(boy) = 0.51 and P(girl) = 0.49. Theory or Experiment?

  4. Myths about chance behavior: • Short-run regularity: Example 4, p. 350 and Example 5, p. 351. • The surprise meeting: Example 6, p. 351 and Example 7, p. 352. • The law of averages: Example 8, p. 353 and gambler’s misconceptions (pp. 352-353)

  5. Personal Probability • A personal probability of an outcome is a number between 0 and 1 that expresses an individual’s judgment of how likely the outcome is. (p. 354) • Can be based on past information or simply someone’s intuition. Examples: • What’s the probability that the Gamecocks will win the next game? • What’s the chance that it will rain tomorrow?

  6. Probability and risk Example 9 on p. 355 • Death from automobile driving relatively more likely than death from cancer due to asbestos exposure. • People tend to worry more about the less likely event.

  7. Chapter 18: Probability Models • A probability model for a random phenomenon describes all possible outcomes and says how to assign probabilities to any collection of outcomes. (p. 363) Properties of a Probability Model: 1. The probability of any event must be a number between 0 and 1. (Rule A in text) 2. If we assign a probability to every possible outcome, the sum of these probabilities must equal one. (Rule B in text)

  8. The Complement Rule (Rule C in text): • Let A be an event. The eventA is the set of all outcomes that are possible, but are not in A. • Two such events are said to be complementary. We say,A is the complement of A. • The probability that an event does not occur is 1 minus the probability that the event does occur. That is, P(A ) = 1 – P(A)

  9. Addition Rule for Mutually Exclusive Events (Rule D in text): • Mutually exclusive events have no outcomes in common. • If two events are mutually exclusive, the probability that one or the other occurs is the sum of their individual probabilities. • If A and B are mutually exclusive, then P(A or B) = P(A) + P(B).

  10. Example: Die Rolling • There are six outcomes, and each is equally likely to appear. • The probability of any one outcome is 1/6. xP(x)1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6

  11. Questions About One Die Roll 1. What is the probability of tossing at most a four? 2. What is the probability of tossing an even number? Complement Rule in the Die Rolling Example: 4. Let A = the die lands on an even number. Then,A = the die lands on an odd number. Find P(A ). 5. Let B = the die lands on at most 4. The complement of B,B, is the die lands on more than 4. Find P(B ).

  12. Example using personal probability: You estimate your chance of getting an A in statistics class to be 0.50 (personal probability). You estimate your chance of getting a B in statistics class to be 0.30. • What is the chance that you will get either an A or a B? • What is the chance that you will earn a C or lower?

  13. Figure 18.1 on p. 356: 36 equally likely outcomes when rolling two dice.

  14. Questions on Roll of Two Dice What is the probability that, when you roll two dice, • the sum of the dice equals five? • there are no sixes? • there is exactly one 6? • there are exactly 2 sixes? • there is at least one six?

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