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Quiz Thursday

Quiz Thursday. Video 1 (Hans Rosling) First 40 Pages of Flatland 3 multiple choice questions at end of hour placed on overhead. Critical Math. Ideas for projects http://www.ams.org/mathmoments. Combinations.

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Quiz Thursday

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  1. Quiz Thursday • Video 1 (Hans Rosling) • First 40 Pages of Flatland • 3 multiple choice questions at end of hour placed on overhead

  2. Critical Math • Ideas for projects • http://www.ams.org/mathmoments

  3. Combinations • The number of was of choosing a team of 7 from among a total group of 30 has shorthand C(30,7) • A long (but not for a calculator) formula would be

  4. Describing Data and Making Predictions Guessing Population Size

  5. Guessing Population Size Suppose the size of a population of fish is 1000 and we tagged 100. If we did another catch of 60 what would be the chance 4 are tagged? Using our notation we can count the number of ways of choosing 4 tagged from 100 and the remaining 56 of second catch from 900 untagged.

  6. Calculation for Fishing • We divide this by all ways of choosing 60 fish from 1000. • Using notation C(100,4)*C(900,56) is the number of ways to do this. • C(1000,60) is the total possible number of second catches • C(100,4)*C(900,56)/C(1000,60)=.13 -Done on a TI calculator-

  7. What Chance 5 tagged? 6 tagged? 7 tagged? • All calculations would be about same • C(100,5)*C(900,55)/C(1000,60)=.17 • C(100,6)*C(900,54)/C(1000,60)=.175 • C(100,7)*C(900,53)/C(1000,60)=.15 • Getting smaller and smaller for 8,9,… • All calculations can be done on calculators

  8. From Calculation We See • If 1000 fish and 10% are tagged (100) and we recatch 60 then the most likely number of tagged to be recaught is 6. • 6/60 = 10% also! • Without doing calculations we could also show this true • In fact this is true independent of population size, percent tagged, number recaught

  9. Biologist Proportion Problem If we knew the size of a population N of animals. Then catch-tag-release-catch strategy will have the most likely number on second catch according to proportions. Tagged/Population ~ tagged on 2nd catch/total caught 2nd catch

  10. What if we don’t population size? • Algebra doesn’t care! So if we didn’t know that the population was XXXX but we tagged 100, re-caught 60 of which 6 were tagged we should guess • 100/XXXX = 6/60 Now we know what volunteers were doing.

  11. More Volunteers • Simple Problems • If we tagged 40 fish, did a recatch of 100 and 7 are tagged about how big do we think the population is going to be? • 40/? = 7/100 so 40*100 = 7*? • 4000/7 is about 570.

  12. More Examples • Unknown population, tag 50 and recatch 200 of which 9 are tagged? • 50/N = 9/200 so 200*50 = 9*N and • N is about 1111 How about the volunteers and goldfish?

  13. Assumptions • The number of fish stays same between catches • The tagged are mixed into the population • We can decide if a fish is tagged • The numbers are large so that a random thing is still close to nonrandom

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