Probability Basics — How to Calculate Probability

Grade: 7-8 | Topic: Statistics

What You Will Learn

After this lesson you will understand what probability is, how to calculate it using the basic formula, and how to express it as a fraction, decimal, or percentage. You will work through examples involving coins, dice, cards, and real-life scenarios, and learn the difference between theoretical and experimental probability.

Theory

What Is Probability?

Probability measures how likely an event is to happen. It is a number between 0 and 1 (inclusive):

• $P = 0$ means the event is impossible (it will never happen)
• $P = 1$ means the event is certain (it will always happen)
• $P = 0.5$ means the event is equally likely to happen or not happen

The Basic Probability Formula

For any event where all outcomes are equally likely:

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

Key terms:

• Experiment — an action that produces results (rolling a die, flipping a coin)
• Outcome — one possible result of the experiment
• Sample space — the set of all possible outcomes
• Event — a specific outcome or set of outcomes you are interested in
• Favorable outcomes — the outcomes that match the event

Quick example: Rolling a standard die and getting an even number.

• Sample space: $\{1, 2, 3, 4, 5, 6\}$ — 6 outcomes total
• Favorable outcomes (even): $\{2, 4, 6\}$ — 3 outcomes

$P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5 = 50\%$

Expressing Probability

Probability can be written in three equivalent forms:

Form	Example
Fraction	$\frac{1}{4}$
Decimal	$0.25$
Percentage	$25\%$

To convert: fraction $\rightarrow$ decimal (divide), decimal $\rightarrow$ percentage (multiply by 100).

Complement of an Event

The complement of event $A$ is "everything that is NOT event $A$." Their probabilities always add up to 1:

$P(\text{not } A) = 1 - P(A)$

If the probability of rain tomorrow is 0.3, the probability of no rain is $1 - 0.3 = 0.7$.

Theoretical vs Experimental Probability

Theoretical probability is what you calculate using the formula — it assumes perfect conditions.

Experimental probability is what you observe by actually doing the experiment:

$P_{\text{experimental}}(\text{event}) = \frac{\text{Number of times event occurred}}{\text{Total number of trials}}$

As you run more trials, experimental probability tends to get closer to the theoretical probability. This is called the Law of Large Numbers.

Worked Examples

Example 1: Flipping a Coin (Easy)

Problem: What is the probability of flipping a fair coin and getting heads?

Step 1: Identify the sample space: $\{\text{Heads}, \text{Tails}\}$ — 2 outcomes.

Step 2: Count favorable outcomes: Heads — 1 outcome.

Step 3: Apply the formula:

$P(\text{Heads}) = \frac{1}{2} = 0.5 = 50\%$

Answer: $P(\text{Heads}) = \frac{1}{2}$ or 50%.

Example 2: Rolling a Die (Easy)

Problem: What is the probability of rolling a number greater than 4 on a standard six-sided die?

Step 1: Sample space: $\{1, 2, 3, 4, 5, 6\}$ — 6 outcomes.

Step 2: Favorable outcomes (greater than 4): $\{5, 6\}$ — 2 outcomes.

Step 3:

$P(\text{greater than } 4) = \frac{2}{6} = \frac{1}{3} \approx 0.333 = 33.3\%$

Answer: $P(\text{greater than 4}) = \frac{1}{3}$ or about 33.3%.

Example 3: Drawing a Card (Medium)

Problem: A standard deck has 52 cards (4 suits, 13 ranks per suit). What is the probability of drawing a heart?

Step 1: Total outcomes: 52 cards.

Step 2: Favorable outcomes: 13 hearts (one of four suits).

Step 3:

$P(\text{heart}) = \frac{13}{52} = \frac{1}{4} = 0.25 = 25\%$

Answer: $P(\text{heart}) = \frac{1}{4}$ or 25%.

Example 4: Using the Complement (Medium)

Problem: A bag has 3 red, 5 blue, and 2 green marbles. What is the probability of NOT drawing a red marble?

Step 1: Total marbles: $3 + 5 + 2 = 10$.

Step 2: Find $P(\text{red})$:

$P(\text{red}) = \frac{3}{10}$

Step 3: Use the complement:

$P(\text{not red}) = 1 - P(\text{red}) = 1 - \frac{3}{10} = \frac{7}{10} = 0.7 = 70\%$

Answer: $P(\text{not red}) = \frac{7}{10}$ or 70%.

Example 5: Experimental Probability (Challenging)

Problem: A student flipped a coin 80 times and got heads 34 times. (a) What is the experimental probability of heads? (b) How does it compare to the theoretical probability? (c) What would you expect after 1,000 flips?

Step 1 — Experimental probability:

$P_{\text{exp}}(\text{Heads}) = \frac{34}{80} = \frac{17}{40} = 0.425 = 42.5\%$

Step 2 — Comparison:

The theoretical probability is $\frac{1}{2} = 50\%$. The experimental result of 42.5% is close but not exactly 50% — this is normal with a limited number of trials.

Step 3 — Prediction:

With 1,000 flips, you would expect the experimental probability to be much closer to 50% (the Law of Large Numbers). You would expect approximately 500 heads, though the exact number will vary.

Answer: (a) Experimental probability $= 42.5\%$. (b) It is 7.5 percentage points below the theoretical 50%. (c) With more flips, the experimental probability approaches 50%.

Common Mistakes

Mistake 1: Probability greater than 1

❌ "I rolled a die. The probability of getting 1 through 6 is $\frac{6}{6} + \frac{1}{6} = \frac{7}{6}$."

✅ Probability can never exceed 1. The probability of getting any number from 1-6 is $\frac{6}{6} = 1$ (certain). Do not add extra events beyond the sample space.

Why this matters: If your answer is greater than 1 (or greater than 100%), you have made an error. This is a useful self-check.

Mistake 2: Forgetting to simplify fractions

❌ $P(\text{heart}) = \frac{13}{52}$ (left unsimplified)

✅ $P(\text{heart}) = \frac{13}{52} = \frac{1}{4}$

Why this matters: While $\frac{13}{52}$ is not wrong, simplified fractions are easier to understand and compare. Most tests expect simplified answers.

Mistake 3: Confusing "at least one" with "exactly one"

A die is rolled. "At least one 5 in two rolls" is different from "exactly one 5 in two rolls."

❌ Treating "at least one" the same as "exactly one."

✅ "At least one" includes getting a 5 on one roll, or both rolls. It is often easier to calculate using the complement: $P(\text{at least one 5}) = 1 - P(\text{no 5 in both rolls})$.

Why this matters: Many word problems use phrases like "at least," "at most," or "exactly." Read carefully and determine which outcomes qualify.

Practice Problems

Try these on your own before checking the answers:

1. A bag contains 4 red, 6 blue, and 5 yellow marbles. What is the probability of drawing a blue marble?
2. What is the probability of rolling an odd number on a standard six-sided die?
3. A standard deck of 52 cards: what is the probability of drawing a king or a queen?
4. If the probability of winning a game is $\frac{2}{7}$, what is the probability of NOT winning?
5. A spinner has 8 equal sections numbered 1-8. What is the probability of landing on a number less than 3?

Click to see answers

1. Total $= 4 + 6 + 5 = 15$. $P(\text{blue}) = \frac{6}{15} = \frac{2}{5} = 40\%$.
2. Odd numbers: $\{1, 3, 5\}$ — 3 out of 6. $P(\text{odd}) = \frac{3}{6} = \frac{1}{2} = 50\%$.
3. Kings: 4, Queens: 4, total favorable: 8. $P(\text{king or queen}) = \frac{8}{52} = \frac{2}{13} \approx 15.4\%$.
4. $P(\text{not winning}) = 1 - \frac{2}{7} = \frac{5}{7} \approx 71.4\%$.
5. Numbers less than 3: $\{1, 2\}$ — 2 outcomes. $P = \frac{2}{8} = \frac{1}{4} = 25\%$.

Summary

• Probability measures likelihood on a scale from 0 (impossible) to 1 (certain).
• Basic formula: $P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$.
• The complement rule: $P(\text{not } A) = 1 - P(A)$.
• Probability can be expressed as a fraction, decimal, or percentage.
• Experimental probability comes from actual trials; theoretical probability comes from logic. With more trials, experimental results approach theoretical values.

Related Topics

• Statistics Basics — Mean, Median, Mode, and Data Analysis
• Mean, Median, Mode, and Range — How to Find Each One
• Data Collection and Analysis — Tables, Charts, and Surveys

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