- Construct probability models
- Compute probabilities of equally likely outcomes
- Compute probabilities of the union of two events
- Use the complement rule to find probabilities
- Compute probability using counting theory
Probability
Many events in life are inherently uncertain: will it snow tomorrow? Am I going to get an ‘A’ in this course? None of these questions can be answered with certainty, however, we might say that some are unlikely, and others are more likely.
Suppose we roll a six-sided number cube. Rolling a number cube is an example of an experiment, or an activity with an observable result. The numbers on the cube are possible results, or outcomes, of this experiment. The set of all possible outcomes of an experiment is called the sample space of the experiment. The sample space for this experiment is [latex]\left\{1,2,3,4,5,6\right\}[/latex]. An event is any subset of a sample space.
The likelihood of an event is known as probability. The probability of an event [latex]p[/latex] is a number that always satisfies [latex]0\le p\le 1[/latex], where [latex]0[/latex] indicates an impossible event and [latex]1[/latex] indicates a certain event. A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. For instance, if there is a [latex]1\%[/latex] chance of winning a raffle and a [latex]99\%[/latex] chance of losing the raffle, a probability model would look much like the table below.
| Outcome | Probability |
|---|---|
| Winning the raffle | [latex]1\%[/latex] |
| Losing the raffle | [latex]99\%[/latex] |
The sum of the probabilities listed in a probability model must equal [latex]1[/latex], or [latex]100\%[/latex].
probability
The probability of an event is a description of how likely it is that an event will happen.
A probability is a number between [latex]0[/latex] and [latex]1[/latex] (that is, between [latex]0\%[/latex] and [latex]100\%[/latex]), where probabilities closer to [latex]100\%[/latex] are very likely to occur, and probabilities closer to [latex]0\%[/latex] are very unlikely to occur. A probability of [latex]0\%[/latex] means the event is impossible, and a probability of [latex]100\%[/latex] means the event will certainly occur.
To calculate the probability of an event, we divide the number of possible outcomes of the event by the number of possible outcomes of the sample space.
[latex]P(\text{outcome}) = \dfrac{\text{Number of ways that outcome can occur}}{\text{Total number of outcomes}}[/latex]
- It is important to note that in order to use this formula, all outcomes must be equally likely to happen.
A probability model is a mathematical description of an experiment listing all possible outcomes and their associated probabilities. It is defined by its sample space, events within the sample space, and probabilities associated with each event.
- The sample space [latex]S[/latex] for a probability model is the set of all possible outcomes.
- An event [latex]A[/latex] is a subset of the sample space [latex]S[/latex].
- Identify every outcome.
- Determine the total number of possible outcomes.
- Compare each outcome to the total number of possible outcomes.