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# Classical probability

## by Ken Kretsch on Oct 11, 2010

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## Classical probabilityPresentation 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
• 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)
• 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:
• 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