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Understanding
Probability

Learn to describe probability as a number from \(0\) to \(1\) and connect it to long-run relative frequency.

Pre-Algebra

What Is Probability?

Probability measures how likely an event is to happen. It is always a number from \(0\) to \(1\), where \(0\) means impossible and \(1\) means certain.

You can also express probability as a fraction or a percentage: \(\frac{1}{2}\) or \(50\%\). All three forms describe the same likelihood.

The Probability Scale

0 0.25 0.5 0.75 1 Impossible Unlikely Equally Likely Likely Certain

Why Does Probability Matter?

★ Weather
A \(70\%\) chance of rain means if similar conditions occur many times, it rains about 70 out of 100 days.
★ Medicine
A vaccine with \(95\%\) effectiveness means about 95 of every 100 people are protected.
★ Games
Understanding probability helps you evaluate risk and make smarter decisions in any game of chance.

Probability appears everywhere: science, sports, business, and daily life. It gives us a common language to talk about uncertainty.

The Probability Formula

$$P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

Experimental Probability: Relative Frequency

Relative frequency is the fraction of times an event actually occurs during an experiment:

$$\text{Relative Frequency} = \frac{\text{times event occurred}}{\text{total trials}}$$

The Long-Run Connection

As you repeat an experiment more and more times, the relative frequency gets closer and closer to the true probability. This idea is called the Law of Large Numbers.

Number of Flips Relative Freq. 0 0.5 1.0 0 100 500 1000 P = 0.5 Observed frequency

Theoretical vs. Experimental

Theoretical Experimental
Based on Mathematical reasoning Actual data from trials
Formula \(\frac{\text{favorable}}{\text{total}}\) \(\frac{\text{times occurred}}{\text{trials}}\)
Example \(P(\text{heads})=0.5\) \(\frac{47}{100}=0.47\)
Precision Exact Approximate (improves with more trials)

How to Calculate Probability

1
Identify the event. What outcome are you looking for?
2
Count favorable outcomes. How many ways can the event happen?
3
Count total outcomes. How many equally likely results are possible?
4
Divide. Compute \(P = \frac{\text{favorable}}{\text{total}}\) and simplify.
5
Check. Is your answer between \(0\) and \(1\)? If not, recount.

Example 1: Rolling a Die

A fair six-sided die has faces \(\{1, 2, 3, 4, 5, 6\}\). What is the probability of rolling an even number?

1 2 3 4 5 6 Highlighted = favorable outcomes

Example 2: Spinner

A spinner is divided into 8 equal sections numbered \(1\) through \(8\). Three sections are colored red. What is the probability of landing on red?

1 2 3 4 5 6 7 8

Example 3: Drawing a Card

A standard deck has 52 cards: 4 suits with 13 cards each. What is the probability of drawing a heart?

♥ 13 cards ♦ 13 cards ♣ 13 cards ♠ 13 cards

The Complement Rule

The complement of event \(A\), written \(A'\), is "everything that is NOT \(A\)." Their probabilities always add to \(1\):

$$P(A) + P(A') = 1$$

Impossible, Certain, and Between

\(P = 0\)
The event cannot happen. Example: rolling a \(7\) on a standard die.
\(0 < P < 1\)
The event might happen. Most real-world probabilities live here.
\(P = 1\)
The event must happen. Example: drawing a card that is red or black.

Common Mistakes

✕ Confusing Fractions and Odds
Probability is a fraction of total outcomes, not a ratio of "for vs. against." \(\frac{1}{6}\) is NOT the same as \(1:5\) odds.
✕ Forgetting Equal Likelihood
The formula only works when outcomes are equally likely. A bent coin doesn't give \(P(\text{heads}) = 0.5\).
✕ Probability > 1 or < 0
If your answer is negative or greater than \(1\), you likely swapped the numerator and denominator or miscounted.

Summary at a Glance

Concept Key Idea Formula / Value
Probability range Always between 0 and 1 \(0 \le P(A) \le 1\)
Theoretical probability Based on reasoning \(\frac{\text{favorable}}{\text{total}}\)
Experimental probability Based on data \(\frac{\text{occurred}}{\text{trials}}\)
Long-run connection More trials = closer match Law of Large Numbers
Complement What doesn't happen \(P(A') = 1 - P(A)\)

Question 1 of 5

A bag contains 3 red, 5 blue, and 2 green marbles. What is the probability of drawing a blue marble?

Question 2 of 5

A coin is flipped 200 times and lands on heads 112 times. What is the experimental probability of heads?

Question 3 of 5

If the probability of snow tomorrow is \(0.15\), what is the probability that it will not snow?

Question 4 of 5

You roll a fair die 6000 times. About how many times would you expect to roll a \(5\)?

Question 5 of 5

Which statement best describes the Law of Large Numbers?