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Welcome to our seventh seminar! We’ll begin shortly Definitions Experiment: an act or operation for the purpose of discovering something unknown. Outcome: the results of an experiment Events: subsets of the outcomes of an experiment Empirical probability: The number of times an event is an outcome P(E) = The number of times the experiment is run This is used when the theoretical probability cannot be calculated (we'll talk about that shortly) Empirical probability The number of times an event is an outcome P(E) = The number of times the experiment is run Since the top number is always either smaller than or equal bottom number, P(E) will always be a number between 0 and 1. It may be expressed as a fraction or a decimal Example A coin is tossed 50 times and it comes up heads 29 of those times. What is empirical probability of flipping a head? 29 P(head) = = 0.58 50 What is the probability of flipping a tail? Tail = 50 - 29 = 21 21 P(tail) = = 0.42 50 Note that P(head) + P(tail) = 1 0.58 + 0.42 = 1 Another Students at KU were polled about their favorite search engines. Here are the results. Google 63 Dogpile 29 Yahoo 34 Bing 24 If one KU student was randomly chosen to do a search on the internet, what is the probablity that the student will choose: Google? Dogpile? Yahoo? Bing? (continued..) First find the total number of "experiments" Total = 63 + 29 + 34 + 24 = 150 63 P(google) = = 0.42 150 29 P(dogpile) = = 0.19 (rounded) 150 34 P(yahoo) = = 0.23 (rounded) 150 24 P(bing) = = 0.16 150 Note that the sum of these is 1: 0.42 + 0.19 + 0.23 + 0.16 = 1 A few more definitions Equally likely outcomes: each of the outcomes of an experiment has the same chance of occurring Theoretical probability (equally likely outcomes): number of outcomes favorable to E Total number of possible outcomes Note that this is different from empirical probablity because it does not require an experiment. P(E) = Law of large numbers: When the number of ‘experiments is very large the empirical probability is the same as the theoretical probability. Most common example Dice Dice have 6 possible outcomes: 1,2,3,4,5,6 Total number of outcomes = 6 What is the probability of rolling a 4? 1 P(4) = 6 What is the probablity of rolling an odd number? (1,3,5) there are 3 odd numbers 3 1 P(odd) = = 6 2 Hints • If an event cannot occur P(E) = 0 (such as rolling a 9 on the dice) • If a probability must occur P(E) = 1 (such as flipping a double headed coin) • 0 ≤ P(E) ≤ 1 P(E) = 1 The sum of all event probablities is 1 The probability that an event will occur and not occur is 1 P(occur) + P(not occur) = 1 so: 1 - P(occur) = P(not occur) Example (deck of cards) Data for a deck of cards: Total cards: 52 # 7's (or any number) = 4 # hearts, clubs, spades, diamonds = 13 # jacks, queens, kings, aces = 4 What is the probablity of selecting an ace? 4 1 P(ace) = = 52 13 What is the probability of not selecting an ace? 1 P (not ace) = 1 13 13 1 12 P (not ace) = 13 13 13 What is the probability of selecting a heart? 13 1 P(heart) = = 52 4 What is the probability of selecting a number card? Total # = 4(2 through 10) = 4(9) = 36 36 9 P(number) = = 52 13 What is the probability of selecting a card between 5 and 9? Total = 4(6 through 8) = 4(3) = 12 12 3 P(5-9) = = 52 13 Odds against The odds against an event occuring is the probablity that the event will not occur divided by the probability that the event will occur P(not occur) failure Odds against = = P(occur) success Example What is the odds against drawing a queen from a deck of card? 4 1 P(queen) = = 52 13 1 P(not queen) = 1 13 13 1 12 P(not queen) = = 13 13 13 12 P(not queen) Odds against = = 13 note this is division: 1 P(queen) 13 12 1 Odds against = 13 13 12 13 Odds against = 13 1 12 Odds against = 1 Odds in favor P(event) success = P(not event) failure What is the odds in favor of drawing a queen from a deck of card? 4 1 P(queen) = = 52 13 1 P(not queen) = 1 13 13 1 12 P(not queen) = = 13 13 13 1 1 12 Odds in favor = 13 12 13 13 13 1 13 1 Odds in favor = = 13 12 12 1 Note: Odds in favor = Odds against Odds in favor = Example If the odds against of a cat being a patient at a vets office is 7 to 2, what is the probability? P(not cat) 7 Odd against = = P(cat) 2 P(cat) = 2 Total odds = 7 + 2 = 9 2 P(cat) = 9 Expected value: used to determine probability over the long term (investments etc.) E = Pi A Where P is the probability of an event occuring and A is the net amount or loss if that event occurs. E = P1A1 + P2 A 2 .......Pn A n One hundred raffle tickets are sold for 2$ each. The grand prize is 50$ and two 20$ prizes are consolation prizes . What is the expected gain? Probability for each ticket winning the grand 1 2 prize is and the consolation prize is 100 100 97 and for winning no prize is 100 1 2 97 E= (48) + (18) + ( 2) 100 100 100 48 36 194 E= + 100 100 100 48 + 36 - 194 E= 100 110 E== -1.10 100 The expected value of each ticket is - $1.10 Fair price = expected value – cost to play this is the ‘break even’ price For the previous example: expected value = -1.10 Cost to play = 2.00 Fair price = -1.10 + 2.00 = 0.90 Each ticket should cost $.72 for the expected value to be zero Note: 1 2 97 (50 .90) + (20 .90) + (.90) 100 100 100 49.1 38.2 87.3 E= 100 E=0 E= Tree diagrams Counting principle: If the first experiment can be done M ways and a second can be done N ways, then the two experiments MN can be done M*N ways. Barney has three pairs of jeans and three shirts to choose from. M = 3, N = 3 so MN = 3*3 = 9 There are 9 possible outcomes. If we have a box with two red, two green and two white balls in it, and we choose two balls without looking, what is the probability of getting two balls of the same color? There are 9 possible outcomes.. Note that 3 of these outcomes are balls of the same color 3 1 P(RR,WW,GG) = = 9 3 Or and And problems “Or” problems have a successful outcome for at least one of the events “And” problems have a favorable outcome for each of the events P (A or B) = P(A) +P(B) - P(A + B) (addition formula) The probability that there will be at least on successful outcome is the sum probability of the first event and second event occurring minus the probablity that both will occur P(A and B) = P(A) P(B) The probability that all outcomes with be favorable is the product of probablities of each event occurring. If I roll a dice, what is the probability that the outcome will be 3 or an even number? Let A be the probability of a 3 Let B be the probability of an even faced dice P (A or B) = P(A) +P(B) - P(A + B) (addition formula) 1 P(A) = 6 3 1 P(B) = = (2,4,6 are the possible dice faces) 6 2 P(A + B) = 0 (you cannot have a 3 that is even) 1 1 P(A or B) = + - 0 6 2 1 3 P(A or B) = + 6 6 4 2 P(A or B) = 6 3 If we have a box with two red, two green and two white balls in it, and we select two balls one at a time what is the probability that the first ball will be red and then the second ball will be red? Let A be the first ball and B be the second ball 2 1 P(A) = 6 3 1 P(B) = 5 P(A + B) = P(A) P(B) P(A + B) = 1 1 1 5 3 15 Thank you for attending!