What Does "Probably" Mean? An Introduction to Probability

Learning Objectives

• Understand basic counting methods.
• Interpret events as probability.
• Understand the sample space.
• Tie events and probability with the sample space.

1. Ahmed and the Truth Behind "Probably"

Ahmed noticed something interesting about his friend Khaled.

No matter what people asked him—

“Are you coming?”
“Did you finish?”
“Will you help?”

—Khaled almost always answered:

“Probably.”

Ahmed got curious. What does "probably" really mean when Khaled says it?

For three weeks, Ahmed kept track every time Khaled said "probably" and whether he actually followed through.

Here’s what he found:

• Khaled said "probably" 30 times.
• He did the thing 21 times.
• He didn’t do it 9 times.

Ahmed did the math:

$21 ÷ 30 = 0.7$

This means that about 70% of the time, when Khaled says "probably," he actually does the thing.
So we might say:
P(Khaled does it | says "probably") = 0.7

What do you think will happen the next time someone asks:

“Are you bringing your laptop tomorrow?”

We use probability in many situations in our lives. Even subconsciously sometimes! Here are some examples of where we use it:

• Weather forecasts: ``There is a 70% chance of rain tomorrow.''
• Games of chance: Rolling dice, drawing cards, or spinning a wheel.
• Medical testing: ``There is a 95% chance the test correctly detects the disease.''
• Sports: What is the chance your team wins the next game?
• Finance: What is the risk that a stock price goes down?
• Traffic What is the average time it will take you to get to school?

2. Why Study Probability?

Probability is the mathematics of chance. It helps us make better choices, analyze risks, and solve real-world problems.

What is probability?

Probability is a number between 0 and 1 that tells us how likely an event is to occur:

• 0 means the event is impossible.
• 1 means the event is certain.
• 0.5 means the event is equally likely to happen or not.
  These probabilities help us make educated guesses, even when outcomes are uncertain.

Examples:

• Tossing a fair coin:
  P(heads) = 1/2

• Rolling a fair six-sided die:
  P(rolling a 4 or 2) = 2/6

• Pulling a red card out of a 52-card deck:
  P(pulling a red card) = 26/52

3. Random Experiments and Events

Random Experiment

A random experiment is an action or process that results in one outcome from a set of possible outcomes, but the exact result cannot be predicted in advance.

Event

A specific outcome or group of outcomes from a random experiment.

Examples:

• Experiment: Toss a coin
  Event: Getting heads

• Experiment: Roll a die
  Event: Rolling an even number = {2, 4, 6}

4. Sample Space and Events

Sample Space (S)

The set of all possible outcomes of an experiment.

Examples:

• Coin toss:
  S = {Heads, Tails}

• Die roll:
  S = {1, 2, 3, 4, 5, 6}

• Drawing a card:
  S = {All 52 cards}

Event (E)

A subset of the sample space.

Quick Tip:

• The action you do → Experiment
• The result you care about → Event
• All possible results → Sample Space

5. Probability of an Event

If all outcomes are equally likely, then:
the probability of an event E is:

$$P(E) = \frac{\text{Number of outcomes in } E}{\text{Total number of outcomes in } S}$$

Example:

• Roll a die. What’s the probability of getting an even number?

Even numbers: {2, 4, 6} → 3 outcomes
Total outcomes: 6

$$P(\text{even}) = \frac{3}{6} = 0.5$$