Probability

Sandy Reed                     Robert Fulton School
                               5300 S. Hermitage
                               Chicago IL 60609
                               (312) 535-9000

Objective: 

The students in grade levels 5-8 will learn, discover, explore and tabulate a 
table of the number of times a specific event can occur.  In this activity, 
students consider the results of two types of events.  One involves repeated 
sampling with replacement; the other, without replacement.  In the first case, 
the events are independent; in the second case, they are dependent. 

Materials Needed:

Groups of two will need:
1.  3 Paper bags
2.  Penny or any coin
3.  3 sets of five chips numbered 1, 2, 3, 4, 5
4.  Paper/Pencil

Strategy:

1.  The class will be divided into groups of two.  Each group will be given
    3 paper bags.  In one bag labeled A, there will be 5 chips numbered 1-5
    and a penny.  In the second bag labeled A, there will be 5 chips numbered
    1-5.  In the third bag labeled B, there will be 5 chips numbered 1-5.
2.  One person in each group will be asked to remove the penny from the bag 
    labeled A and place the three bags to the side.  One person in each group
    will look at the penny to determine the total number of sides of that penny.
3.  After determining the number of sides of a penny, the group will flip the 
    penny once to determine the likelihood of getting a head.
4.  The instructor will write the definition of probability and the formula for
    probability.
5.  The group will flip the coin twice and write a table of the total number of 
    outcomes that can occur when flipping a coin twice while the instructor
    writes the table on the overhead projector.  The instructor will show the 
    students how to write the table in a uniformed pattern.  For example, 
    HH, HT, TT, TH, beginning with the heads as starting the pattern.
6.  The student will use their probability formula to determine that 
    P=1/2x1/2=1/4 with each 1/2 being each flip (2 flips) and the probability
    of flipping two heads is 1 out of 4 or 1/4.  
7.  The group will flip the coin 3 times and determine the table of 3 flips
    using their own pattern and also, write the probability of 3 flips.
8.  The group will then place the two bags labeled A on their desk and place the 
    penny to the side.  The student will need to determine the table for all of 
    the total possible outcomes that can occur from drawing 5 chips in each of 
    the two bags and replacing the chips.  The group will take turns drawing 2 
    chips (1 chip from one bag and 1 chip from the other bag) and writing down 
    the results.  The group will continue to replace the chips in the bag after 
    the 2 chips are drawn.  The group will need to determine that there are 25 
    total possible outcomes that can occur with replacement in independent 
    sampling. 
9.  The group will then place the one bag labeled B, draw 2 times, and write 
    down the results of the first and second draw.  The group will not replace 
    the chip from the first draw in the bag.  For example, if the first draw was 
    a 2, the 2 would not be replaced in the bag.  The student will need to 
    determine the table for all the total possible outcomes that can occur from
    drawing 5 chips from one bag and not replacing the chip from the first draw.
    The group will need to determine that there are only 20 total possible
    outcomes that can occur without replacement in dependent sampling.