What is Probability?

The study of the chance associated with the occurrence of events.

Two Types of Probability

   Two types of approaches to probability can be classified as the classical approach and the relative frequency approach.  The third type is called subjective.  For grades 3-5, the Tennessee State Standards focus is on the classical approach.  For grades 6-8, the Tennessee Standards include the relative frequency approach.

Classical Probability 

Rolling dice or tossing a coin are activities associated with a classical approach to probability.  In these cases, you can list all the possible outcomes of an experiment and determine the actual probabilities of each simple event.

Listing All Possible Outcomes of a Probabilistic Experiment

There are various ways to list all possible outcomes of an experiment.  One way is to use a tree diagram.  Since this is prescribed by the Tennessee Standards, we will use this method.

         

Tree Diagrams

Tree diagrams are an ordered approach to listing all possible outcomes.  Such a technique helps ensure that you do not miss any possibilities.              

3 Children Example:  A couple wants to have exactly 3 children.  Assume that each child is either a Boy or a Girl and that there are no duplicate births.  List all possible orderings for the three children.   

 

 

 

 

 

 

 

  Probability of Single Event with Equally Likely Outcomes   

Term:  Sample Space - the list of all possible outcomes from a probabilistic experiment.  Each individual item in the list is called a Simple Event or Single Event.

If we can list all possible different outcomes of an experiment and each outcome is equally likely, then the probability of any one outcome is 1 divided by the total number of outcomes.  
 
A simple example of this is flipping a coin.  If you flip a fair coin one time, then there are two different possible outcomes (Head shows or Tail shows).  Each outcome is equally likely since the coin is fair.  Thus the probability of any one of the distinct outcomes (say "Heads showing") is 1/2  (1 divided by the total number of outcomes.)  The total probability associated with the sample space is 1  (1/n added together n times).

Example:   A couple wants to have exactly 3 children.  Assume that the chance of a boy or girl is equally likely at each birth.  
What is the probability that they will have exactly 3 girls? 
What is the probability that they will have exactly 3 boys?

 

 

 

 

 

 

 

Probability of Compound Events that can be Written as a Combination of Single Events with Equally Likely Outcomes

If an event can be written as a combination of distinct items from the sample space (i.e. Single Events) then you can find the probability of the event by adding the individual probabilities.

        2-Coin Example:  Consider an experiment where you flip a fair coin twice.  
        List all possible outcomes.  ________________________  
        Is each outcome equally likely?  ______  
        What is the probability of each outcome?  _____  
        Find the probability that exactly 2 tails show.  ________  

Now, consider the probability that a exactly one tail shows.  Exactly one tail showing in two flips is actually an example of a compound event.  It is the combination of two simple events  (Tails-Heads or Heads-Tails) that each have a probability of 1/4.  Thus the Probability of exactly one tail is the same as the probability of either T-H or H-T which equals 1/4 + 1/4 = 1/2.

        Find the probability of at least 1 tail showing.  _______
        Find the probability of at most 1 tail showing.  ________

 

3-Children Example:  A couple wants to have exactly 3 children.  Assume that the chance of a boy or girl is equally likely at each birth.  Find the following probabilities:
        exactly 2 girls   _______
        exactly 2 boys   _______
        at least 2 boys   _______
        at most 2 boys  _______
        at least 1 girl    _______
        at most 1 girl    _______

Probability Problems from your Textbooks

    4th Grade:  Pages 446-447.  Spinner problems, probability of equally likely events
    5th Grade:  Pages 468-469 & 476-477.  More spinner problems.  Determining simple probability

A Spinner Question:  Consider a spinner that is divided equally with two colors Red and Blue. Player A and B take turns spinning the spinner.  If the two players get the same color then player A wins.  If the two players get different results, then player B wins.  Is this a fair game?  Explain.

 

 

Relative Frequency Probabilities (Grades 6-8 Standards)

A Lot of Fun for Students in a lot of different grades!!!
        Rolling Dice
        Flipping Coins
        Drawing from Bags without Looking (i.e. Sampling)
        Sampling with M&M's  (http://m-ms.com/cai/mms/faq.html#what_percent lists the percent of each color in different types of M&M candies.)

 

Example:  How to Win at Wheel of Fortune?  Answer:  Use Relative Frequency Probabilities.

(Source. "Teaching Middle School Mathematics Activities, Materials and Problems. by Krulik and Rudnick. page 161.  Allyn & Bacon, Boston. 2000.)

In your class it would be fun to play a round of Wheel of Fortune.  Then talk to the class about the best way to guess the letters if you go first to increase your chances of getting a letter in the words.  Have the students guess (hypothesize) which letters appear most frequently in words.  Write several of their answers on the board.  Then talk about a strategy for trying to find out which letters would have the best chance of appearing most often in a phrase.  

Have each student take a paragraph from a newspaper or their math book or any other course and practice tallying the relative frequency for each letter in the alphabet.  You can record your tallies below.

 

Letter Place a tally mark here each time the letter occurs in your paragraph Total Number of Tallies for the given letter
A    
B    
C    
D    
E    
F    
G    
H    
I    
J    
K    
L    
M    
N    
O    
P    
Q    
R    
S    
T    
U    
V    
W    
X    
Y    
Z    
  TOTAL (add the items in the right hand column)  

Combine the results for the entire class on the board.  Determine which letters occurred more frequently.  See if the hypotheses that the students formed before collecting the data were correct.  You can talk about the relative frequency as the probability that such a letter would occur.  Let the students record their strategy for selecting the letters when playing Wheel of Fortune.  Then play another round of Wheel of Fortune and see if the strategy works.
        

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