## Probability

A measure of the likelihood that a specific event will occur is the probability of that event. Before determining the probability of an event, students must be able to find the total number of possible outcomes. The total number of possible outcomes can be found by making an organized list, by making a tree diagram, or by using multiplication. If there are m possible choices, or outcomes, for one experiment or problem and n possible choices or outcomes for a second experiment or problem, then the total number of possible choices or outcomes is m × n. If you have 3 sizes of shirts and 2 choices of color, there are two possible colors for each of the three sizes, which means there are six possible outcomes because 3 × 2 = 6. Multiplication can be used when you only need to find the total number of possible outcomes. This is known as the fundamental counting principle.

The result of an experiment is called an outcome. If the experiment is to toss a 1−6 number cube, then there are six possible outcomes, one for each face of the cube. An event is any collection of outcomes. Examples of events for tossing a number cube are that the number tossed is even, that the number is 1 or 2, or that the number is 3. The probability of an event is a measure of the likelihood that an event will occur. The probability is always a number between 0 and 1. A probability of 0 means that an event is impossible while a probability of 1 means that an event is certain.

When a coin is tossed, there are two outcomes, heads or tails. Either outcome is equally likely. When a 1−6 number cube is tossed, each face is equally likely to turn up. When a marble is chosen from a bag of thoroughly mixed marbles of the same size, without looking, each marble has the same chance of being chosen. Such outcomes are said to be equally likely outcomes. Sometimes the word fair is used when the outcomes are equally likely, as in “fair coin” or “fair number cube.” To indicate that each outcome is equally likely, the word random is used, such as in saying “The object is chosen at random.”

If each outcome is equally likely, the theoretical probability of an event is the ratio of the number of outcomes in the event to the total number of possible outcomes. At this grade level, students will determine simple probability by finding the ratio of favorable outcomes to all outcomes.

Teaching Model 23.4: Problem-Solving Strategy: Make an Organized List