Basic Probability Concepts

Probability is the mathematical way of describing how likely an event is to happen. Whether you are flipping a coin, rolling a die, or predicting the weather, probability helps us measure certainty.

The Probability Scale

Probability is always expressed as a number between $0$ and $1$. It can be written as a fraction, a decimal, or a percentage.

• $0$ (Impossible): The event cannot happen under any circumstances.
• $1$ (Certain): The event is guaranteed to happen.
• $0.5$ or $\frac{1}{2}$ (Equally Likely): There is a 50-50 chance of the event occurring.

How to Calculate Theoretical Probability

To find the theoretical probability of an event, you compare the number of ways that specific event can happen to the total number of possible outcomes.

The formula is:

$P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Note: "Favorable outcomes" just means the outcomes you are looking for in the problem!

Example Problems

Example 1: Rolling a Die

What is the probability of rolling an even number on a standard die?

1. Find the total possible outcomes: A standard die has 6 sides, so the possible outcomes are $1, 2, 3, 4, 5, 6$. (Total = $6$)
2. Find the favorable outcomes: The even numbers on a die are $2, 4, 6$. (Favorable = $3$)
3. Apply the formula: $P(\text{Even}) = \frac{3}{6}$
4. Simplify: $P(\text{Even}) = \frac{1}{2}$

So, the probability is $\frac{1}{2}$, $0.5$, or $50\%$.

Example 2: Drawing Marbles

A bag has 3 red and 2 blue marbles. What is the probability of drawing a red marble?

1. Find the total possible outcomes: Add the marbles together. $3 \text{ red} + 2 \text{ blue} = 5 \text{ total marbles}$.
2. Find the favorable outcomes: We want a red marble, and there are $3$ red marbles.
3. Apply the formula: $P(\text{Red}) = \frac{3}{5}$

The probability of drawing a red marble is $\frac{3}{5}$, which can also be written as $0.6$ or $60\%$.