1 / 13

Chapter 6: Probability : The Study of Randomness

Chapter 6: Probability : The Study of Randomness. “We figured the odds as best we could, and then we rolled the dice.” US President Jimmy Carter June 10, 1976. 6.1 Randomness (p. 310-316). Random phenomenon An outcome that cannot be predicted

kevyn-maxwell
Download Presentation

Chapter 6: Probability : The Study of Randomness

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 6:Probability : The Study of Randomness “We figured the odds as best we could, and then we rolled the dice.” US President Jimmy Carter June 10, 1976

  2. 6.1 Randomness (p. 310-316) • Random phenomenon • An outcome that cannot be predicted • Has a regular distribution over many repetitions • Probability • Proportion of times that an event occurs in many repeated events of a random phenomenon • Independent events (trials) • Outcome of an event (trial) does not influence the outcome of any other event (trial)

  3. Independent event Rolling a single die twice Result on second roll is independent of the first Event that is not Independent Picking two cards from a deck (one at a time with no replacement) If you pick a card and do not replace it and reshuffle the deck, then the probability of a red card on the second pick is dependent upon what the first card happened to be. Examples:

  4. 6.2 Probability Models (P. 317-337) It is often important and necessary to provide a mathematical description or model for randomness. • Sample space • The set of all possible outcomes of a random phenomenon

  5. Example of a Sample Space • Consider the SUM obtained when two dice are rolled. • Create a table to display the sums that can be obtained for this event. • The sample space contains _______ outcomes.

  6. Probability Rules • If A is an event, then the probability P(A) is a number between o and 1, inclusive. • P(A does not happen) = 1 – P(A) • The complement of A • Two events are DISJOINT of they have no outcomes in common. • If A and B are disjoint, then P(A or B) = P(A) + P(B) • Sometimes the phrase MUTUALLY EXCLUSIVE is used to describe disjoint events

  7. Probability Rules: An Example • In a roll of two dice • Suppose A = rolling a sum of 7 and B = rolling a sum of 12, the P(A or B) = 6/36 + 1/36 = 7/36 (A and B are disjoint). • Suppose C = rolling a sum of 7 and D = rolling an odd sum, the P(C or D) = P(C) + P(D) – P(C and D) = 6/36 + 18/36 – 6/36 = 18/36 = ½ (C and D are NOT DISJOINT since 7 is an odd number)

  8. Probability Rules • Two events are INDEPENDENT if knowing that one occurs does not change the probability of the other. • If events A and B are independent, then P(A and B) = P(A) X P(B)

  9. Examples: • Suppose A = getting a head on a first toss and B = getting a head on a second toss. A and B are independent. • P(A and B) = (1/2) X (1/2) = ¼ • Suppose C = getting a red card by picking a card from a randomly shuffled deck of cards and D = getting a red card by picking a second card from the deck in which the first card WAS NOT REPLACED. C and D are NOT independent. • P(C) = 26/52. • If the first card is red, then P(C and D) = (26/52) X (25/51). • If the first card had been placed back into the deck then • P(C and D) = (26/52) X (26/52) because C and D ARE independent.

  10. Probability formulas are useful, however, sometimes they are not needed if sample spaces are small and you use some common sense. • Consider a family of two children. There are four possible boy/girl combinations, ALL EQUALLY LIKELY. • P(at least one child is a girl) = • P(two girls) = • P(two girls| one child is a girl) = • P(two girls| oldest child is a girl) = • P(exactly one boy | one child is a girl) = • P(exactly one boy | youngest child is a girl) =

  11. Now suppose that a family unknown to you has two children. • If one of the children is a boy, what is the probability that the other child is a boy? • If the oldest child is a boy, what is the probability that the other child is a boy? Many intelligent people think that 50% is the answer to both of the above. You should be able to show that this is NOT the case.

  12. More about Probability (P. 341-355) • There are basic laws that govern wise and efficient use of probability. • FOR ANY TWO EVENTS A and B, • P(A or B) = P(A) + P(B) – P(A and B) • P(A and B) = 0 if A and B are disjoint • P(B|A) = P(A and B)/P(A)

  13. Examples Using the Sum of Two Dice • P(sum = 7 or sum = 11) = • P(sum = 7 and sum = 11) = • P(sum = 7 or at least one die shows a 5) = • P(faces show same number or sum>9) = • P(sum = 6 | one die show a 4) = • P(sum is even | sum>9) =

More Related