Classical probability
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Classical probability

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Classical probability Classical probability Presentation Transcript

  • Probability
    • Quantifying the likelihood that something is going to happen.
    • A number from 0 to 1, inclusive
      • 0 - Impossible
      • 1 - Certain, guaranteed
      • ½ - a “toss up”
    • Can be expressed as a fraction (in lowest terms), decimal, or percent
      • Usually starts out as a fraction
  • Probability definition: Event
    • An event is one occurrence of the activity whose probability is being calculated.
      • E.g., we are calculating the probability of dice, an event is one roll of the dice.
    • A simple event cannot be broken down into smaller components
      • Rolling one dice is a simple event
    • A compound event is made up of several simple events
      • The probability of a compound event is usually a function of the component simple events.
      • Rolling two dice is a compound event.
  • Probability definitions: Outcome, sample space
    • An outcome is one possible result of the event.
      • Rolling a five is one possible outcome of rolling one dice
      • Rolling a seven is one possible outcome of rolling two dice
    • The sample space is the list of all possible outcomes
      • One dice: 1, 2, 3, 4, 5, or 6
      • Two dice: See next slide
    • The size of the sample space is the total number of possible outcomes
      • One dice: sample space size is 6
      • Two dice: sample space size is 36
    • A success is an outcome that we want to measure
    • A failure is an outcome that we do not want to measure
      • Failures = Sample space – successes
  • Two Dice Sample Space First Die 1 2 3 4 5 6 2 nd Die 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12
  • Probability Symbols and Calculation
    • The letter P denotes a probability.
    • Capital letters (A, B, C, etc) represent outcomes
    • P(A) denotes the probability of outcome A occurring
    • Where a success is when outcome A occurs
  • For example: One Dice
    • What is the probability of rolling a five with one dice?
      • Sample space: 1 2 3 4 5 or 6
      • Sample space size: 6
      • Successful rolls:
      • Number of successes:
      • P(5) =
    • What is the probability of rolling an odd number?
      • Successful rolls:
      • Number of successes:
      • P(Prime) =
  • For example: Two Dice
    • What is the probability of rolling a five with one dice?
      • Sample space size: 36
      • Successful rolls:
      • Number of successes:
      • P(5) =
    • What is the probability of rolling a prime number?
      • Number of successes:
      • P(Prime) =
  • Types of Probability
    • Classical
      • AKA Theoretical or Empirical
      • Events and outcomes in sample space can be determined from the ‘rules of the game’
      • E.g., Wheel of fortune
    • Geometric
      • Sample space is some area, a successful outcome is hitting some target
    • Experimental
      • AKA Relative frequency
      • Some activity is observed
      • Sample space size is the total number of events observed
      • Success is the subset of events in which out outcome occurred
      • E.g., basketball toss
  • Classical probability: Coin flip
    • Event: coin flip
    • Sample space: heads or tails
    • Sample space size: 2
    • Probability of flipping heads
    • Sucesses:
    • # of Successes
    • P(Heads)
  • Classical Probability: Cards
    • Event: drawing one (or more) cards
    • Sample space: a deck cards, two colors, each color has two suits, each suit has 13 ranks deuce to ten, three face cards, ace
    • Sample Space size: 52
    • What is the probability of drawing a 10 of spades?
    • Successes:
    • Number of successes:
    • P(10 ♠)
  • Classic Classical Probability: Cards Successes # of success P P(Jack) P(Red) P(Heart)
  • Your turn
    • From a deck of cards
    • P(Face card) =
    • P(Red ace) =
    • P(6 or less) =
  • Classical Probability: Collections
    • Sample space: a set of items of different characteristics
      • Sample space size. We will know the total and numbers of each characteristics
    • Event: Picking one (or more) items with a specific characteristics
    • E.g., A box of balls: 4 red, 2 blue, 2 green, 2 yellow, 1 white and 1 black.
    • Sample size:
    • P(red)
      • Number of successes:
    • P(Black or white)
      • Number of successes:
  • Your Turn
    • If all the tokens we in a bag and picked at random:
    • P(Square)
    • P(2)
    • P(3 in a triangle)
    1 3 2 1 3 2 1 2 3 1 1 2 1 1 1 2 1 1 1 2 3 3
  • Classic Classical Probability
    • Collections with multiple characteristics
    • P(North) =
    • P(Junior) =
    • P(South upperclassman) =
    Frosh Soph Junior Senior North 400 375 325 350 South 350 300 325 275
  • Classic Classical Probability
    • Collections with multiple characteristics
    • P(North) =
    • P(Junior) =
    • P(South upperclassman) =
    Frosh Soph Junior Senior North 400 375 325 350 South 350 300 325 275
  • Classical Probability: Spinner
    • Event: Spinning the wheel
    • Outcome: Spinner stops at a space
    • Sample space: individual spaces
    • Sample space size: # of spaces
    • P(1)
    • P(red) =
    • P(Prime)
    1 3 2 4
  • Do now
    • A wheel of fortune has 15 spaces and costs 25 cents to play. If you win, you get a $3 prize
    • Another wheel has 10 spaces and also costs 25 cents. If you win, you get a prize worth $2.25.
    • If you were down to your last 25 cents, which wheel would you play?
    • If you had 10 dollars to spend (25 cents at a time), which wheel would you play?
  • Identifying the events and sample space
    • Sometimes we have to enumerate the sample space.
    • How many ways are there to arrange the genders of three children?
    • Sample space size?
  • Questions, always questions
    • What is the probability of having three girls?
    • P(one boy)?
    • P(Youngest is a boy)?
    • P(At least one boy)?
  • More types of probability
    • Geometric probability
    • The event is hitting a target on some surface.
  • Complimentary events
    • If A represents the occurrence of an event, then Ā represents the event not occurring.
    • Ā is the compliment of A
    • P( Ā ) = 1 – P(A)
  • Odds
    • Odds against are the ratio P( Ā ):P(A), reduced to lowest terms
    • Odds in favor are the reciprocal of the odds against
    • What are the odds
      • Against drawing a red card
      • In favor of drawing an ace
      • Against rolling a 5
  • Odds
    • Payoff odds against: Net profit : Amount bet
    • Example: roulette wheel
    • The payoff odds for picking one number are 35:1
      • If you bet $1, you win $35, plus your original bet.
      • How much do you win if you bet $5?
    • What are the actual odds?
      • 38 spots on the wheel
    • Casinos are profitable because the payoff odds are less than the actual odds