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Glencoe Algebra 2

PT: Experimental vs. Theoretical , Law of Large Numbers, Replacement, Sample Space & Size

RW: Candy

Materials needed: enough small non-transparent bags of differently colored candy for each member of the class.

Step 1: Have each student open her or his bag of candy, but NOT look inside.  Have each student take out one piece of candy, record its color and then place it back into the bag.  Shake the bag of candy to mix the one you chose in with the other pieces.

Step 2:  Repeat (Step 1) 20 times and record your findings.

Step 3: Have each student find the experimental probability of each color of candy in their bag.  Then have the class find the experimental probability of each color for the whole class.

Step 4: Have each student pour out the contents of their bags and determine the actual number of each color they have in their candy bag.  Then have the student find the theoretical probability of each color in groups of four.  Then find the theoretical probability of each color for the entire class.

Questions for students:

1) Compare the experimental to the theoretical probabilities for:

            -You individual probabilities

            -Your ‘group of four’ probabilities

            -Your class’s probabilities

Which pair of probabilities was closer to each other?

Why do you think this is the case?

2) How different do you think the results would have been if every student in the class dumped their individual bags into their own bowl and conducted the same experiment?

Glencoe Algebra 1, p. 190-192, 227

PT: Inference, Experiments, Randomness

RW: Fishing

Materials: paper lunch bag, two different color beans, and a paper cup

Proportions, percents, and probabilities

Imagine that you are asked to determine the number of fish in a nearby pond.  To count the fish one by one, you could remove the fish from the pond and stack them to one side, or mark each fish so you would not count them over and over again.  Counting like this could be hazardous to a fish’s health.

To determine the number of animals in a population, scientists often use the capture-recapture method.  A number of animals are captured, carefully tagged, and returned to their native habitat. Then a second group of animals is captured and counted, and the number of tagged animals is noted.  Scientists then use proportions to estimate the number in the entire population. 

In this investigation, you will work in pairs to model the process used by scientists to estimate the number of fish in a large lake.  The lunch bag will represent the lake, the beans will represent fish, and the paper cup will represent a net. 

Step 1: Make a chart like the one above on a sheet of paper, numbering the casting column up to 5.

Step 2: Write your name on the paper bag.  Empty one bag of beans into the bag.

Step 3: Use your net, a 5 oz paper cup, to remove a sample of fish.  Count the number of fish you netted.  Since this number will remain constant for all casts, you can record this number in each row column A.

Step 4: Replace all the beans you counted with beans of a different color.  Put these "tagged fish" back into the lake and shake gently to mix the fish.

Step 5: Use the net, 5 oz paper cup, to remove a sample of fish.  Count the total number of fish in your sample and record this number in column B.  Then count the number of tagged fish in this sample and record this number in column C of your chart.  Return these fish into the lake and shake again to mix the fish.

Step 6: Cast your "net" a second time and record your findings.  Continue casting and recording until you have counted five samples.

Estimating the population:

a)  Write a proportion that relates the numbers in your chart and the estimated number of fish in the lake.

b)  Imagine that the only information you had was the data you recorded after your first casting.  What would have been your estimate of the number of fish in the lake?  Justify your reasoning.  Record your estimate in the last column of your chart.

c)  Make estimates for each of your casting and record them in the last column of your chart.

d)   Taking all of your castings into consideration, if you had to give an official estimate of the number of fish in the lake, what would it be?  Justify your reasoning.

Count all the fish in your lake.

Record the actual and estimated populations on your class chart.

a)  How close were your estimates?

b)  How would you account for the difference between your estimates and the actual number of fish in the lake?

c)  What percentage of the fish in the lake was tagged?

d)  What percentage of the fish in your sample was tagged?

e)  Suppose you went fishing at that lake and caught one fish.  What is the probability, or chance, that the fish you caught would be tagged? Justify your reasoning.

Write a paragraph or two that relates proportions, percents, and probability in this situation.

Glencoe Algebra 2, p. 767

RW: Games

In this project, you will design and construct a game based on probability.  All games have certain features in common.  The object of the game is to win, so your game should have a starting point and an ending point.  But not all games involve a playing board.  Some games involve playing with dice, others with a spinner, and still others with marbles or cards.

Follow these steps to design and construct you own probability game.

1)  Brainstorm with your group to choose the features your game will have.

2)  Outline a plan you can follow.

3)  Use materials you can easily find.  It’s okay to borrow dice, spinners, marbles, cards and so on from other games.  Be original; don’t design your game so it looks like another game that already exists.

4)  Carry out your plan.

5)  Determine the probabilities of the events, which will take place in your game.  (You might want to make the event that have less probability of happening worth more points than events with a large probability).

6)  Play your game several times.  Keep tract of strategies used to win the game.

7)  Write several paragraphs describing how you designed and play your game, the rules of the game, and how someone can win the game.

Key to Problem Bank: