Basic Probability

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Interpret probability as a ratio of favorable outcomes to total outcomes.
  • Compute simple probabilities for coins, dice, cards, and everyday situations.
  • Use complements to solve problems efficiently.
  • Distinguish theoretical vs. experimental probability.

Key Ideas

Probability of an event:

\[ P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} \]

  • Probabilities range from 0 to 1.
  • Complement rule: \[ P(\text{not A}) = 1 - P(A) \]
Category Count
Vanilla 12
Chocolate 8
Strawberry 5
Total 25

This table represents the number of students who prefer each ice-cream flavor.
Since probability is defined as “favorable outcomes divided by total outcomes,”
each count can be turned into a probability by dividing by the total of 25.

For example:

  • 8 students prefer chocolate, so
    \[ P(\text{chocolate}) = \frac{8}{25}. \]

Tables like this appear often in probability problems because they allow you to read counts directly and convert them into probabilities quickly.

Common Problem Types

Basic Ratio Probability

Count favorable outcomes and divide by total.

Example:
Rolling a die → \(P(\text{rolling a 4}) = 1/6\).

Probability From a Table or List

Use given frequencies.

Example:
Out of 20 students, 8 prefer chocolate → \(8/20 = 0.4\).

Complement Probability

Find probability of “at least one,” “not this,” or “not that.”

Example:
\(P(\text{not heads}) = 1 - 1/2 = 1/2\).

Probability With Cards (basic)

Know total outcomes = 52.

Example:
\(P(\text{drawing a heart}) = 13/52 = 1/4\).

Interpreting Experimental Probability

Based on observed data.

Example:
If an event occurs 15 out of 60 trials → probability ≈ 0.25.

Strategies

  • Always identify total outcomes first.
  • Use complements when “at least one” or “none” is involved.
  • Reduce fractions when possible.
  • If unsure: write or imagine the sample space.

Worked Examples

Example 1

A coin is flipped. What is \(P(\text{heads})\)?

\[ 1/2 \]

Example 2

A bag has 3 red, 5 blue, 2 green. Probability of blue?

\[ 5/10 = 1/2 \]

Example 3

From a deck, probability of drawing a spade?

\[ 13/52 = 1/4 \]

WarningCommon Mistakes
  • Using total outcomes incorrectly (especially with cards).
  • Forgetting that probability is always between 0 and 1.
  • Ignoring complements when they simplify the problem.
  • Mixing up theoretical and experimental probabilities.

Practice Problems

  1. Roll a die. What is \(P(2)\)?
  2. A jar has 4 red, 6 yellow marbles. Probability of yellow?
  3. A card is drawn from 52. Probability it’s a face card?
  4. If 9 out of 30 students walk to school, what is the experimental probability?
  1. \(1/6\)
  2. \(6/10 = 3/5\)
  3. Face cards = 12 → \(12/52 = 3/13\)
  4. \(9/30 = 3/10\)

Summary

  • Probability = favorable ÷ total.
  • Complements help with “not” or “at least one.”
  • Reduce fractions and interpret results in context.
  • Draw a quick model if needed.
  • If the problem says “random,” assume equal likelihood.
  • Complement rule often saves major time.