#### MAIN IDEAS

One might well summarize most or all of what should go on in probability in K to 8 classrooms as follows: The conduct of experiments followed by age-appropriate analysis of and reflection on the results.

With the splitting of this chapter from the previous one on data analysis, new activities have been added and what I hope you will find is a more careful development of probability ideas over the K-8 grade span.

#### The Probability Continuum

Perhaps the biggest probability idea for the early grades is the notion of chance. In this chapter an effort is made o help students see how the chance of a future event occurring exists on a continuum from impossible to certain. Placing numerical values from 0 to 1 on this continuum is the process of assigning a probability to an event.
#### Theoretical and Experimental Probability

Three definitions of probability are used. In the most general sense, probability is a measure of the chance of a future event occurring. How this measure is arrived at - through objective analysis or through experimentation - gives rise to the two related definitions of theoretical and experimental probability respectively. Although an experimental probability may be easy to compute, the related concept of the Law of Large Numbers is a bit more illusive since it is based on an intuitive concept of limit. The experimental probability approaches the actual probability as the number of trials increases. For an infinite number of trials, the two would be exactly equal.
#### Computing Theoretical Probability: Sample Spaces, Independent and Dependent Events

For simple, single-stage events, a theoretical probability is relatively simple to understand. When an experiment or probabilistic situation involves two or more stages, confusion sets in. The first difficulty is with the size of the sample space. For example, when rolling two dice, why are there 36 elements in the sample space rather than just 11? When the events in the multistage experiment are dependent, the complexity increases.

No doubt you will experience the same difficulty with your students that they will experience with school children.
#### Simulations

Drawing colored cubes from a bag, rolling dice, and spinning spinners are necessary for helping students develop appropriate concepts of probability. However, simulations of real-world situations show us why probability is important in the first place. Simulations are also a great example of the manner in which mathematics can be used to model the real world, make sense of real phenomena, and use this analysis to make important predictions.