Activity 1: Drawing three marbles from six


 

1.1. First hands-on activity



Supplies: marbles, a brown paper bag, and a tally sheet

Choose six non-rolling marbles, 3 of one color, 2 of another, and 1 of a third.  You have a brown paper bag.  Place the marbles in the bag, close it and shake it, and draw out (without looking) three at once.  Record the number of each color you got on your score sheet.  (Below you can see that there are six possible outcomes.)  Put the marbles back in the bag, close it, and shake it.  Again draw out three at once without looking, and record the colors.  Continue this 50 times.  Tally your scores.  


 

1.2.  Computing the probability

Suppose you have 6 marbles, 3 red, 2 blue, and 1 clear, in your bag.  If you draw out three of the six marbles randomly, what are the chances of getting each of the six combinations?

When we choose three out of six, we need Pascal’s triangle up to six:

                                                                        1

                                                               1                   1

                                                     1                   2                 1

                                             1             3                      3             1

                                    1            4                    6                  4            1

                             1             5              10                   10           5               1

                       1            6         15                   20              15             6              1

 

In order to compute the values of Pascal’s triangle, use the nCr notation, which you find on the TI-83/84 under MATH, PROB, 3:nCr.  You may read it as, “the number of ways one can choose r items out of a collection of n items”, or simply “n choose r”.

 

                                                                        0c0     

                                                               1c0              1c1

                                                     2c0               2c1             2c2

                                            3c0            3c1                3c2           3c3

                                    4c0            4c1               4c2              4c3          4c4

                             5c0         5c1             5c2                5c3           5c4          5c5

                     6c0             6c1       6c2                 6c3              6c4          6c5           6c6

 

So, for example, 4c2 = 6 means I can choose 2 items out of a collection of 4 items in 6 different ways. 

Let’s check that this is true. Name these four items A, B, C and D. You can choose: A and B, A and C, A and D, B and C, B and D, or C and D. The order in which you choose the items is irrelevant. When choosing A and B you may select first A and  then B. or first B and then A, or pick up both at the same time.

List all the ways you can choose three different letters out of A, B, C, D, E and F, in order to show that the total number of ways you can choose 3 objects out of 6 is really 20 (which is written 6c3=20). 

There are six different possible outcomes of choosing three marbles out of your bag which contains 6 marbles: R, R, R, B, B, and C. They are RRR, RRB, RRC, RBB, RBC, and BBC.  Let’s compute the probabilities of getting each of the six.

             red        blue      clear

RRR:    (3c3) * (2c0) * (1c0)  =  1*1*1=1 out of 20 =   5%

RRB:    (3c2) * (2c1) * (1c0)  =  3*2*1=6 out of 20 = 30%

RRC:    (3c2) * (2c0) * (1c1)  =  3*1*1=3 out of 20 = 15%

RBB:    (3c1) * (2c2) * (1c0)  =  3*1*1=3 out of 20 = 15%

RBC:    (3c1) * (2c1) * (1c1)  =  3*2*1=6 out of 20 = 30%

BBC:    (3c0) * (2c2) * (1c1)  =  1*1*1=1 out of 20 =   5%

                                                         20 out of 20 = 100%

Let’s check that the number of ways of choosing two red marbles and one blue marble, RRB, from the bag is 6.

 Marbles in the bag:  R  R  R  B  B  C

Choices:                  R  R      B

                              R  R          B

                              R      R  B

                              R      R      B

                                  R  R  B

                                  R  R      B


 

 (Click image for printable version)


Return to Drawing Marbles Index

 


Webpage implementation by Kristina Brantley