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. 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. 1000 Fish 100 Tagged 4 on 2 nd Catch Tagged 96 Not Recaught but Tagged 900 Untagged 56 on 2 nd Catch Untagged 900-56=844 Never Caught

7. 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-

8. 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

9. 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

10. 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 2 nd catch/total caught 2 nd catch

11. 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.

12. 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.

13. 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?

14. 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