Counting Techniques and Expressions

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• Misc
  • Math Olympiad
    • Regional college math contest
    • Saturday, Feb. 7, 9:30 - 12:30
    • See Prof. Towsley for more info and to sign up
• Questions?
• Symbolic Math in Matlab
  • Interacts poorly with numeric features
  • Don’t use it until you understand numeric Matlab, and if it’s the only thing you’re doing in that session
  • Use via mupad command if you do use it, to keep symbolic and numeric environments separate
• Counting and Expectation
  • Multiplication principle
    • Independent events, e.g., dials on lock
    • Total # choices is product of individual choices
    • (Example: how many 8-bit patterns are there?)
    • Another example: pairs of cards
      • 12 face cards
      • 40 non-face cards
      • = 12 × 40 = 480 possible pairs with one face card and one non-face card
  • Combinations
    • Using combinations to count ways event can occur
    • Event = selecting r things from n
      • No replacement
      • Order doesn’t matter
    • e.g., horses in race, how many different top-three choices
    • nCr = n! / (n-r)! r!
    • Example: number of ways to choose 2 out 12 face cards
      • 12! / 2! 10!
      • = (12)(11) / 2
      • = 66
  • Expected value
    • Outcomes (values) with probabilities
    • Add products of outcome values times probabilities to get “expected value” (average)
    • Example: expected value of picking from non-face cards
      • Outcomes = 1, 2, 3, …, 10
      • Each with probability 1/10
      • E(X) = 1 (1/10) + 2(1/10) + … + 10(1/10)
        • = (1/10)( 1+2+3+…+10)
        • = (1/10) (55)
        • = 5.5
• Variables and Expressions Lab
  • Expected value for lottery is about -18 cents
• Next
  • Algorithms
  • Watch “What Is An ‘Algorithm’?” video at https://cloud.ensemblevideo.com/app/sites/index.aspx?destinationID=IlO-D-b-vUWVkAsj2Is5Zg
  • Or read section 3.1

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