Probability

Moss, Laura Willis              South Shore Community Academy
9628 South Peoria Street        312-933-8180
Chicago, Illinois  60643

Objectives:

The students will derive a general formula for the probability of a 
given event given each possible outcome is equally likely. 

Apparatus Needed:
                          
6 dice
30 colored balls (1 each of five different colors)
6 coins
6 paper plate holders
6 plastic cups
6 paper bags
19 ping pong balls (numbered from 0 to 18)

Recommended strategy:

The students were told they had an opportunity to win a prize in 
today's class by playing the lottery.  They were to pick a 3 digit 
number, a 4 digit number (repeats are allowed) and a combination of 4 
numbers from 1 to 18 (no number could be chosen twice). 

The students were divided into groups of 3 or 4.  Each group received 
a paper bag containing 1 die, 5 colored balls (1 of each  color-blue, 
green, yellow, orange and pink), 1 coin, 1 paper plate holder and 1 
plastic cup.  Each group had to conduct three experiments. 
     1.  Flip the coin 50 times into the paper plate holder, record
         the results (heads or tails).
     2.  Pull a ball from the bag, record its color, replace the
         ball and repeat this process a total of 50 times.
     3.  Use the cup to shake the die.  Roll the die into the paper
         plate holder.  Record the result and repeat the process 
         60 times.

As a class the data from each experiment was collected.  The students 
were asked questions such as:  Were the results of each experiment 
what you would have expected?   What would you expect if we were to 
repeat the process 1000 times?  10,000 times?  10,000,000 times?

The class then generated the formula for the probability of an event P(E)
    P(E) = number of favorable outcomes
            number of possible outcomes

Then the class discussed whether or not the formula will tell us 
exactly what will happen for a given event.  More examples were 
discussed.

The "lottery" ended the lesson.  Students with a knowledge of 
permutations and combinations can calculate the probability of each 
event. 

                     1       1
P(3 digit number) = --- = ----  
                    103    1000

                      1      1
P(4 digit number) = --- = ------
                    104   10,000


P(4 number from 1 to 18) =      1    =     1    
                               18!        3060
                              14!4!