Probability Is a Number Between 0 and 1

From Events to a Probability Scale

In Events as Subsets of the Sample Space, we described an event as a set of outcomes that satisfy a condition. For example, when rolling two dice, “the sum is 8” names an event made up of several ordered pairs. A probability gives a numerical description of how likely an event is according to a chance model.

Probability is measured on a scale from 0 to 1, including both endpoints. In a finite model in which every possible outcome has positive probability, a probability of 0 represents an impossible event and a probability of 1 represents a certain event. In general models, probability 0 means the event has probability zero, not necessarily that it is impossible; probability 1 means probability one, not necessarily logical certainty. Values between 0 and 1 describe events that are possible but not certain. The closer a probability is to 1, the more likely the event is considered under that model. The closer it is to 0, the less likely it is considered.

The scale can be written as decimals, fractions, or percentages. For example, \(0.25\), \(\frac{1}{4}\), and \(25\%\) express the same probability. To convert a decimal probability to a percentage, multiply by 100 and attach the percent sign. To convert a percentage to a decimal, divide by 100.

Here, \(P(A)\) means “the probability of event \(A\).” This inequality is a useful first check: a claimed probability cannot be negative or greater than 1. For instance, \(-0.1\) and \(1.3\) are not valid probabilities. Written as percentages, the same scale runs from 0% to 100%.

What the Endpoints Mean

An event is impossible when it contains no outcome that the stated chance process can produce. In Events as Subsets of the Sample Space, the event “the sum of two six-sided dice is 13” had no outcomes, because the largest possible sum is 12. In the model where the dice are six-sided, the probability of that event is 0.

An event is certain when every possible outcome in the model belongs to it. For the same two dice, the event “the sum is between 2 and 12, inclusive” includes every possible outcome, so its probability is 1. These endpoint values refer to the chance model and the event as stated. Changing the process or the event can change the probability.

For example, if a weather model gives a 2% chance of rain during an outdoor event, rain is unlikely according to that model, but it is not impossible. If a model gives a 95% chance that a delivery arrives by Friday, arrival by Friday is very likely, but it is not guaranteed. The remaining possibility matters: the event could fail to occur.

Interpreting Values Between 0 and 1

A probability between 0 and 1 tells us that an event is neither impossible nor certain under the model. Its decimal value also gives a convenient percentage interpretation. A probability of \(0.02\) is 2%, while a probability of \(0.95\) is 95%. State the event and the setting when interpreting these values; a number without its context is incomplete.

Words such as “unlikely” and “very likely” can help communicate a probability, but they are descriptions, not exact categories with universal cutoffs. A probability of 0.02 indicates a small chance in the specified model. A probability of 0.95 indicates a high chance. Neither number guarantees what will happen on one occasion.

It is also important to distinguish the probability of an event from the event itself. The event is a set of outcomes, such as “the package arrives by Friday.” Its probability is the numerical value the model assigns to that set. The probability is not a prediction that the event must happen, and it does not change the event’s definition.

Compare Probabilities Without Overstating Them

When two events are described by probabilities from appropriate models, their values can be compared on the same scale. For example, \(0.02\) is less than \(0.95\), so an event with probability \(0.02\) is less likely under its model than an event with probability \(0.95\) under its model. The comparison concerns the stated probabilities; it does not establish that the events have the same consequences or arise under the same conditions.

A probability close to 0 is not the same as 0. A probability close to 1 is not the same as 1. Preserve this distinction in words: “unlikely” is not “impossible,” and “very likely” is not “certain.” This precision is especially useful when explaining a model’s prediction to someone making a decision.

A probability can also be expressed as the chance that an event does not occur. If an event has probability 0.95, the remaining probability on the scale is 0.05, or 5%, for the event not occurring. In context, if the room is available with probability 0.95, it is unavailable with probability \(1-0.95=0.05\), assuming those are the only two possibilities in the model. This is one way to check that the interpretation accounts for both possibilities.

Common Mistakes and AP Exam Tips

• Reporting a number outside the scale. A value below 0 or above 1 cannot be a probability. Check the scale before interpreting a reported value.
• Calling a small probability impossible. In a finite model where every possible outcome has positive probability, probability 0 represents an impossible event; in general, an event with probability 0 is not necessarily impossible. A positive value such as 0.02 represents an event that remains possible, though unlikely.
• Calling a high probability certain. Probability 0.95 is high, but it is not 1. Say “very likely” or “95% probability,” not “guaranteed.”
• Confusing a decimal with its percentage. \(0.02\) is 2%, not 0.02%. Multiply a decimal by 100 to convert it to a percentage.
• Giving an interpretation without context. “The probability is 95%” does not identify what event has that probability. Name the event and the situation.
• Turning a model probability into a promise about a particular outcome. A model can assign a high probability to an event that does not occur. Explain what the model says while leaving room for other possible outcomes.

For a full-credit interpretation, state the probability in a useful form, identify the event and context, and describe its likelihood accurately. If the value is between 0 and 1, do not replace “unlikely” with “impossible” or “very likely” with “certain.” Keep the conclusion within what the stated chance model supports.