Range of Probability

The range of probability is a fundamental concept in probability theory. It defines the possible values a probability can take. Probability measures the likelihood of an event occurring, and this likelihood is always bounded between 0 and 1 (inclusive).

This means:

A probability of 0 signifies that an event is impossible.
A probability of 1 signifies that an event is certain to occur.
All other probabilities fall between 0 and 1, representing varying degrees of likelihood.

Formulae

The range of probability is formally expressed as:

$0 \le P(E) \le 1$

Where:

$P(E)$ represents the probability of event $E$.
$0$ represents the impossibility.
$1$ represents the certainty.

Examples

Here are some examples to illustrate the concept:

Example-1: Consider flipping a fair coin.

The probability of getting heads, $P(\text{Heads}) = 0.5$ (or 50%), which is between 0 and 1.
The probability of getting a side that isn’t heads or tails (e.g., the coin standing on its edge), $P(\text{Edge}) = 0$. This is impossible.
The probability of getting either heads or tails, $P(\text{Heads or Tails}) = 1$. This is certain.

Example-2: Consider rolling a standard six-sided die.

The probability of rolling a 3, $P(3) = \frac{1}{6}$, which is between 0 and 1.
The probability of rolling a number greater than 6, $P(\text{>6}) = 0$, which is impossible.
The probability of rolling a number from 1 to 6, $P(1 \le \text{number} \le 6) = 1$, which is certain.

Common mistakes by students

Students often make the following mistakes:

Probabilities greater than 1: Assigning a probability value larger than 1 (e.g., 1.2 or 120%). This indicates a misunderstanding of the probability range.
Negative Probabilities: Assigning negative values to probabilities. This is also mathematically incorrect.
Confusing probabilities with other concepts: Some students might confuse probability with percentages or ratios without understanding the context.

Real Life Application

The range of probability is crucial in many real-life applications:

Weather Forecasting: Meteorologists use probabilities to predict the likelihood of rain, sunshine, or other weather conditions. The forecast always falls between 0% and 100% (or 0 and 1).
Insurance: Insurance companies use probability to calculate the risk of events like accidents or illnesses, which affects the cost of insurance policies.
Sports Analytics: Analyzing the probability of a team winning a game based on player statistics, historical performance, etc.
Medical Diagnosis: Doctors use probability to assess the likelihood of a patient having a particular disease based on symptoms and test results.

Fun Fact

The concept of probability has roots in games of chance, such as gambling. Early mathematicians like Blaise Pascal and Pierre de Fermat developed the foundations of probability theory while analyzing games of dice and cards!