## Probability

At this grade level, students learn to decide whether an event is certain, likely, unlikely, or impossible. They record and display the results of probability experiments, and use the results of experiments to predict outcomes. Students also learn to use probability to decide if a game is fair or unfair.

The result of an experiment is called an outcome. If the experiment is tossing 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 the 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 number cube is tossed, each face is equally likely to turn up. When a marble is chosen from a group of 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, 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.”

The probability of an event is given by the ratio of the number of favorable outcomes to the number of possible outcomes. When you toss a fair coin, the outcomes “heads” and “tails” (denoted by H and T) are equally likely. The set {H, T} represents all possible outcomes and is called the sample space associated with this experiment. The event that the coin comes up heads or tails is denoted by {H} or {T}.

The calculation of the probability of “heads” is based on the fact that the sample space {H, T} consists of two equally likely outcomes. The event {H} consists of one of these outcomes and its probability is calculated as

 P(H) = Number of Outcomes in {H} Number of Outcomes in {H,T}

It is on this basis that we arrive at P(H) = . P(T) = is derived in a similar way.

When you toss a fair coin twice, the outcomes {HH, HT, TH, TT} are equally likely. Here the event "one head" corresponds to the outcomes {HT, TH}, and the probability of this event is or . The outcome “no heads” corresponds to {TT} and has a probability of .

Teaching Model 7.3: Outcomes and Probability