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IGE104: LOGIC AND MATHEMATICS FOR DAILY LIVING 1 Lecture 9: Probability IGE104 3/2010- Lecture 9 April 1, 2011 THE BIG PICTURE QUESTION Why is it that we cannot predict the future? 2 IGE104 3/2010- Lecture 9 April 1, 2011 RANDOMNESS IN DAILY LIFE Whether Car Accidents Course grades How well can we predict these events? 3 IGE104 3/2010- Lecture 9 April 1, 2011 SIMPLE EXAMPLE: COIN TOSS What happens if I toss two coins? List the possible outcomes: Each row in this table is an outcome Which outcome would you expect to occur? Coin A Coin B 4 IGE104 3/2010- Lecture 9 April 1, 2011 SIMPLE EXAMPLE: COIN TOSS We define an event as one or more possible outcomes that all have the same property of interest Coin A Which outcomes belong to the event E = one Head ? What are the chances of having the event E occur? Coin B 5 IGE104 3/2010- Lecture 9 April 1, 2011 THEORETICAL PROBABILITY When all outcomes are equally likely, we say that the probability of an event is Probability (event) = number of outcomes in the event total number outcomes We can use set notation to write down the outcomes in the event. This is an a priori probability because we calculate it before we actually see any events. IGE104 3/2010- Lecture 9 6 April 1, 2011 THEORETIC PROBABILITY Ever time we toss a coin there are two possible outcomes: Heads or Tails If the coin is “fair” then each outcome is equally likely So the probability of tossing one head is just 1 out of 2 Probability(heads) = 1 / 2 7 IGE104 3/2010- Lecture 9 April 1, 2011 WHAT DOES THE NUMBER MEAN? We saw that the probability of tossing one Head is ½, which is the same as 0.5…what does this mean? Well you can think of a probability as a percent. So a probability of ½ is the same as something being 50% likely to occur. What does a probability of zero represent? What does a probability of one represent? 8 IGE104 3/2010- Lecture 9 April 1, 2011 CALCULATING PROBABILITIES WITH EQUALLY LIKELY OUTCOMES If we assume that a person is equally likely to be born in any month of the year, what is the probability that a random person will have a birthday in April? Pr(April birthday) = ? 9 IGE104 3/2010- Lecture 9 April 1, 2011 CALCULATING PROBABILITIES WITH EQUALLY LIKELY OUTCOMES If we assume that a person is equally likely to be born in any dayof the year, what is the probability that a random person will have a birthday in April? Pr(April birthday) = 31 = ~0.08219 365 10 IGE104 3/2010- Lecture 9 April 1, 2011 CALCULATING PROBABILITIES WITH EQUALLY LIKELY OUTCOMES Example: What is the probability that a family with 3 children has 2 girls and one boy? Pr(2 girls and 1 boy)= ? 11 IGE104 3/2010- Lecture 9 April 1, 2011 CALCULATING PROBABILITIES WITH EQUALLY LIKELY OUTCOMES Example: What is the probability that a family with 3 children has 2 girls and one boy? Pr(2 girls and 1 boy)= 3 = 0.375 8 12 IGE104 3/2010- Lecture 9 April 1, 2011 COUNTING: THE MULTIPLICATION RULE How to find to total number of possible outcomes. Previously we had 3 children, where each child could be either a boy or a girl. The total possible number of outcomes was 2 x 2 x 2 = 8 Tree Diagram: Place Diagram: 13 IGE104 3/2010- Lecture 9 April 1, 2011 COUNTING: THE MULTIPLICATION RULE The multiplication rule: the total number of possibilities is just the product of the number of possibilities for each “trial”. Example: A restaurant menu offers two choices for an appetizer, five choices for a main course, and three options for a dessert. How many different three-course meals are possible? 14 IGE104 3/2010- Lecture 9 April 1, 2011 USING THE COUNTING RULE Suppose you roll one die 10 times. What is the probability that you get a 1 on the first roll and a 6 on the next 9 rolls? 610 = 60466176 1/ 610 = 0.000000016538 15 IGE104 3/2010- Lecture 9 April 1, 2011 COUNTING RULE: BE CAREFUL Suppose you toss one coin 10 times. What is the probability that you toss 1 head and 9 tails? 16 IGE104 3/2010- Lecture 9 April 1, 2011 EMPIRICAL PROBABILITY Up until now we have practiced finding the theoretical probability of an event. But what if we want to find the probability of an event that we have already observed in the real world? 17 IGE104 3/2010- Lecture 9 April 1, 2011 EMPIRICAL COIN TOSS Consider an experiment where we toss two-coins, with possible events: 0 heads, 1 head, or 2 heads Suppose we conduct this experiment 100 times (we toss two coins 100 times total) and we get the following results: 0 heads occurs 22 times 1 head occurs 51 times 2 heads occurs 27 times Compared to the theoretical probabilities, do we have reason to believe that the coins are unfair? IGE104 3/2010- Lecture 9 18 April 1, 2011 EXAMPLE: THE 500 YEAR FLOOD Geological records indicate that a river has crested above a particular high flood level at least four times in the past 2,000 years. What is the empirical probability that the river will crest above this flood level next year? 19 IGE104 3/2010- Lecture 9 April 1, 2011 PROBABILITY OF AN EVENT NOT OCCURRING The probability of an event not occurring can be found using the probability that an event does occur. Recall from our set notation the relationship between an event and its negation. So Pr(E) + Pr(not E) = 1 Thus to find Pr(not E), we can just use Pr(not E) = 1 – Pr(E) Not E E 20 IGE104 3/2010- Lecture 9 April 1, 2011 PROBABILITY OF AN EVENT NOT OCCURRING The probability of an event not occurring can be found using the probability that an event does occur. Recall from our set notation the relationship between an event and its negation. So Pr(E) + Pr(not E) = 1 Thus to find Pr(not E), we can just use Pr(not E) = 1 – Pr(E) Not E E 21 IGE104 3/2010- Lecture 9 April 1, 2011 PROBABILITY OF AN EVENT NOT OCCURRING Probability of not drawing an ace from a full shuffled deck of cards. A deck of cards contains the following numbers {2♠, 3♠, 4♠, 5♠, 6♠, 7♠, 8♠, 9♠, 10♠, J♠, Q♠, K♠, A♠, 2♣, 3♣, 4♣, 5♣, 6♣, 7♣, 8♣, 9♣, 10♣, J♣, Q♣, K♣, A♣, 2♥, 3♥, 4♥, 5♥, 6♥, 7♥, 8♥, 9♥, 10♥, J♥, Q♥, K♥, A♥, 2♦, 3♦, 4♦, 5♦, 6♦, 7♦, 8♦, 9♦, 10♦, J♦, Q♦, K♦, A♦} Pr(not an Ace) = 1 – Pr(Ace) = ? 22 IGE104 3/2010- Lecture 9 April 1, 2011 SO FAR WHAT WE HAVE FOUND The probability of an event E with equally likely outcomes is just the number of outcomes that satisfy the event over the number of all possible outcomes Pr(event) = # outcomes in the event total outcomes IGE104 3/2010- Lecture 9 The probability of an impossible event E is Pr(E)=0 The probability of an event E that will definitely occur is Pr(E)=1 The probability of an event E not occurring is 23 Pr(not E) = 1 – Pr(E) April 1, 2011 PROBABILITY QUICK CHECK Can you ever have a probability that is greater than 1? Can you ever have a probability that is less than 0? 24 IGE104 3/2010- Lecture 9 April 1, 2011 PROBABILITY DISTRIBUTIONS: TOSSING 3 COINS Find the probability distribution for the number of heads that occur when three coins are tossed simultaneously. 25 IGE104 3/2010- Lecture 9 April 1, 2011 PROBABILITY DISTRIBUTIONS: SUM OF TWO DICE What is the probability distribution of the possible sums of two simultaneous dice rolls. 26 IGE104 3/2010- Lecture 9 April 1, 2011 BIBLIOGRAPHY (1) Bennett, J. O. , and W. L. Briggs. General Education Mathematics: New Approaches for a New Millennium. AMATYC Review, vol. 21, no. 1, Fall 1999, pp. 3-16. http://math.ucdenver.edu/~wbriggs/qr/AMATYCPaper.html 27 IGE104 3/2010- Lecture 9 April 1, 2011