Classical probability

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Classical probability

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

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