# 5.15 - Probability

The student will

• determine the probability of an outcome by constructing a sample space or using the Fundamental (Basic) Counting Principle.

### BIG IDEAS

• So that I can use probability in sport competitions to try to figure out who is mostly likely to win
• So that a Meteorologist can predict the weather
• So that I can figure out all the different ways that I can arrange an outfit
• So that I can choose a combination of food items on a menu with many options
• So that I can win a variety of board games such as Boggle, Clue, or Scrabble

### UNDERSTANDING THE STANDARD

• A spirit of investigation and experimentation should permeate probability instruction, where students are actively engaged in explorations and have opportunities to use manipulatives, tables, tree diagrams, and lists.
• Probability is the measure of likelihood that an event will occur. An event is a collection of outcomes from an investigation or experiment.
• The probability of an event can be expressed as a fraction, where the numerator represents the number of favorable outcomes and the denominator represents the total number of possible outcomes. If all the outcomes of an event are equally likely to occur, the probability of the event is equal to:
number of favorable outcomes
total number of possible outcomes.

• Probability is quantified as a number between zero and one. An event is “impossible” if it has a probability of zero (e.g., the probability that the month of April will have 31 days).  An event is “certain” if it has a probability of one (e.g., the probability that if today is Thursday then tomorrow will be Friday).
• When a probability experiment has very few trials, the results can be misleading. The more times an experiment is done, the closer the experimental probability comes to the theoretical probability (e.g., a coin lands heads up half of the time).
• Students should have opportunities to describe in informal terms (i.e., impossible, unlikely, equally likely, likely, and certain) the degree of likelihood of an event occurring. Activities should include practical examples.
• A sample space represents all possible outcomes of an experiment. The sample space may be organized in a list, chart, or tree diagram.
• Tree diagrams can be used to illustrate all possible outcomes in a sample space. For example, how many different outfit combinations can you make from two shirts (red and blue) and three pants (black, white, khaki)? The sample space displayed in a tree diagram would show the outfit combinations: red-black; red-white; red-khaki; blue-black; blue-white; blue-khaki.  Exploring the use of tree diagrams for modeling combinations helps students develop the Fundamental Counting Principle.  For this problem, applying the Fundamental Counting Principle shows there are 2 x 3 = 6 outcomes.
• The Fundamental (Basic) Counting Principle is a computational procedure to determine the total number of possible outcomes when there are multiple choices or several events. It is the product of the number of outcomes for each choice or event that can be chosen individually. For example, the possible final outcomes or outfits of four shirts (green, yellow, blue, red), two shorts (tan or black), and three shoes (sneakers, sandals, flip flops) is 4 x 2 x 3 = 24 outfits.

### ESSENTIALS

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

• Construct a sample space, using a tree diagram to identify all possible outcomes.
• Construct a sample space, using a list or chart to represent all possible outcomes.
• Determine the probability of an outcome by constructing a sample space. The sample space will have a total of 24 or fewer equally likely possible outcomes.
• Determine the number of possible outcomes by using the Fundamental (Basic) Counting Principle.

### KEY VOCABULARY

Updated: Mar 06, 2019