Investigating Probability

The focus today was on looking at probability. I looked at basic probability properties and rules for calculation. Many of these were covered last class but I did introduce mutually exclusive events. I also discussed the concept of independence informally. I used roulette to illustrate many of the properties and rules for calculation.

Afterward, I tried an investigation I saw at the NCTM annual conference. You have 3 strings and you are blindly tying ends together. The possible outcomes are to form one large loop, one medium and one small loop, or three small loops. The class actively discussed what the possible outcomes were and how to identify or count the outcomes. As expected, students tried to apply formulas they had learned in the past without considering the problem situation. I used a tree diagram (a new way to represent probability situations) and we worked through the probabilities. The results were counter to the class's intuition as to what result should be most likely. I liked this activity; it illustrated many components of the basic probability properties and rules and helped show how tree diagrams are created and used for calculating probabilities.

Below is the outline of the class along with commentary that is enclosed in square brackets and italicized, [like this].

·         Reference basic probability rules

·         Roulette practice

o   P(red)

o   P(even)

o   P(3^rd 12)

o   P(1-18 or center column) [did not reference general addition rule here, but noted how we accounted for numbers that were both 1-18 and in center column]

o   P(2^nd 12 and 3^rd column)

o   P(black and even)

o   P(black or even) [now referenced general addition rule]

o   P(not 19-36)

o   P(0 or 00)

o   P(0 and 00)

o   P(1^st spin red and 2^nd spin black) [spins are independent so that probabilities are multiplied]

o   Is the event  “8 and black” mutually exclusive

o   Is the event “25 and black” mutually exclusive

·         Strings

o   Take 3 strings

o   Grab and fold in half

o   Swirl to randomize

o   Tie 2 ends together, do it again, tie last pair together

o   What are possible outcomes [as discussed above, outcomes are: 1 loop, 2 loops, or 3 loops]

o   Which do you think most likely to occur, which least likely to occur [class guessed that 2 loops would be most common]

o   Tabulate results and represent [broke student work to have students share thinking on how they were configuring outcomes; needed to do this as some students were getting themselves off track with thoughts like there are 6 ends being matched with each other so 6 x 6 = 36 total outcomes]

     §  What are actual probabilities, show using trees [worked through a branch and then had students attempt to complete other branches, most got stuck]