Probabilities, Counting, and Equally Likely Outcomes - Finite Math
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Probabilities, Counting, and Equally Likely Outcomes - Finite Math

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Finite Math
Probabilities, Counting, and Equally Likely Outcomes

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Probabilities, Counting, and Equally Likely Outcomes - Finite Math Presentation Transcript

  • 1. 2.1Probabilities, Counting, and Equally Likely Outcomes
  • 2. Events An event is a subset of the sample space of an experiment. Ex: The event of an “even number” from the experiment of rolling a die.  Sample space: {1, 2, 3, 4, 5, 6}  “qualifying elements”: {2, 4, 6}
  • 3. Quiz 2.1 #1 Consider the experiment of flipping a coin twice. How many elements are in the event of “flipping at least one head?” (Hint: draw a tree diagram [1.4] and determine the sample space first, then determine which elements “qualify”) A. 2 B. 3 C. 4
  • 4. Quiz 2.1 #1 Consider the experiment of flipping a coin twice. How many elements are in the event of “flipping at least one head?” (Hint: draw a tree diagram [1.4] and determine the sample space first, then determine which elements “qualify”) A. 2 B. 3 C. 4 Answer: B
  • 5. Two Dice “Box” Method Visual representation of the sample space of two dice roll:
  • 6. Two Dice Box Example Suppose we want to know the number of elements in the event “sum less than or equal to 4.” Here’s how we use this box (Circles represent qualifying elements): First die Second die Therefore, the answer is 6.
  • 7. Quiz 2.1 #2 Consider the experiment of rolling two dice. What is the number of elements in the event “difference between rolls is at least 3?” A. 12 B. 14 C. 16
  • 8. Quiz 2.1 #2 Consider the experiment of rolling two dice. What is the number of elements in the event “difference between rolls is at least 3?” A. 12 B. 14 C. 16 Answer: A
  • 9. Outcomes and Probabilities For any experiment, each outcome is said to have a “probability” or “weight” – the likelihood of that event compared to other ones. The probability of all possible outcomes of an experiment must sum up to 1.
  • 10. Equally Likely Outcomes For some experiments, it is intuitive that all outcomes of the experiment are equally likely. For example, the outcomes {1, 2, 3, 4, 5, 6} from rolling a “fair” die is equally likely. Since the probabilities have to sum up to one, each element has a probability of 1/6.
  • 11. Weighted ProbabilitiesLet’s consider the following experiment: An urn has 2 red, 1 white, and 1 blue balls. Let O1 = red, O2 = white, O3 = blue.  O means Outcome Since the chance of drawing each ball is equally likely, each ball has ¼ chance of being drawn w1 = .5, w2 = .25, w3 = .25  W for weights w1 + w2 + w3 = 1
  • 12. Quiz 2.1 #3
  • 13. Quiz 2.1 #3 Let consider an experiment of drawing a card from a deck of cards. What’s the probability of drawing an Ace? A. 1/12 B. 1/13 C. 1/52
  • 14. Quiz 2.1 #3 Let consider an experiment of drawing a card from a deck of cards. What’s the probability of drawing an Ace? A. 1/12 B. 1/13 C. 1/52
  • 15. Quiz 2.1 #3 Let consider an experiment of drawing a card from a deck of cards. What’s the probability of drawing an Ace? A. 1/12 B. 1/13 C. 1/52 Answer: B
  • 16. Summary Definition:  event  outcome, weight How to determine the number of elements in an event How to use “Two Dice Box” Equally likely outcomes  Determining probabilities of events with an experiment containing equally likely outcomes.
  • 17.  Features  27 Recorded Lectures  Over 116 practice problems with recorded solutions  Discussion boards/homework help  Visit finitehelp.com to find out more For special offers and additional content...Follow us on twitter @finitehelp Become a fan on Facebook