Rolling a Die

What is the probability of rolling a $4$ on a standard six-sided die?

Identify favorable outcomes: There is only one face showing $4$ = Favorable outcomes = $1$

Count total possible outcomes: A die has 6 faces: $1, 2, 3, 4, 5, 6$ = Total outcomes = $6$

Apply the formula: $P(4) = \frac{\text{favorable}}{\text{total}} = \frac{1}{6}$ = $P(4) = \frac{1}{6}$

Convert to percentage (optional): $\frac{1}{6} \approx 0.167 \approx 16.7\%$ = About $16.7\%$ chance

Answer: $P(4) = \frac{1}{6} \approx 16.7\%$

Drawing a Red Card

A standard deck has 52 cards (26 red, 26 black). What is the probability of drawing a red card?

Identify favorable outcomes: Red cards = hearts + diamonds = $26$ = Favorable outcomes = $26$

Count total possible outcomes: Total cards in deck = Total outcomes = $52$

Apply the formula: $P(\text{red}) = \frac{26}{52}$ = $P(\text{red}) = \frac{26}{52}$

Simplify the fraction: $\frac{26}{52} = \frac{1}{2}$ = $P(\text{red}) = \frac{1}{2} = 50\%$

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

Rolling an Even Number

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

List all even numbers on a die: Even numbers: $2, 4, 6$ = Favorable outcomes = $3$

Count total possible outcomes: Die has 6 faces = Total outcomes = $6$

Apply the formula: $P(\text{even}) = \frac{3}{6}$ = $P(\text{even}) = \frac{3}{6}$

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

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

Spinner with 8 Sections

A spinner has 8 equal sections numbered $1$ to $8$. What is the probability of landing on a number greater than $5$?

List numbers greater than 5: Numbers $> 5$: $6, 7, 8$ = Favorable outcomes = $3$

Count total sections: Spinner has 8 equal sections = Total outcomes = $8$

Apply the formula: $P(> 5) = \frac{3}{8}$ = $P(> 5) = \frac{3}{8}$

Convert to percentage: $\frac{3}{8} = 0.375 = 37.5\%$ = $37.5\%$ chance

Answer: $P(> 5) = \frac{3}{8} = 37.5\%$

Mistake: Confusing favorable outcomes with total outcomes

Why: Students sometimes put the total on top of the fraction instead of the favorable outcomes.

Correct: Remember: favorable outcomes go on TOP (numerator), total outcomes go on BOTTOM (denominator). $P = \frac{\text{what you want}}{\text{all possibilities}}$

Mistake: Forgetting to simplify fractions

Why: While $\frac{26}{52}$ is correct, the simplified form $\frac{1}{2}$ is clearer and easier to interpret.

Correct: Always simplify probability fractions when possible. $\frac{26}{52} = \frac{1}{2}$

Mistake: Thinking probability can be greater than 1

Why: Some students calculate probabilities like $\frac{8}{5}$ when miscounting outcomes.

Correct: Probability is ALWAYS between $0$ and $1$ (or $0\%$ to $100\%$). If your answer is greater than $1$, recheck your counts!

Mistake: Not counting all favorable outcomes

Why: When asked for 'at least 5', students might only count 5, forgetting 6, 7, 8, etc.

Correct: Read carefully! 'At least 5' means $5, 6, 7, 8...$; 'greater than 5' means $6, 7, 8...$

Weather Forecasting

Meteorologists calculate the probability of rain based on historical data and current conditions.

If rain occurred on 7 out of 20 similar weather days, the probability is $\frac{7}{20} = 35\%$.

Game Show Prizes

Game shows use probability to balance excitement with fair chances of winning.

If a wheel has 20 sections with 3 showing 'Grand Prize', $P(\text{grand prize}) = \frac{3}{20} = 15\%$.

Quality Control

Factories use probability to estimate defect rates and maintain product quality.

If 3 out of 500 items are defective, $P(\text{defect}) = \frac{3}{500} = 0.6\%$.

Q: Why is probability never greater than 1?

A: The number of favorable outcomes can never exceed the total outcomes. At most, every outcome is favorable, giving $P = \frac{n}{n} = 1$ (certainty).

Q: What does a probability of 0.5 mean?

A: A probability of $0.5$ (or $\frac{1}{2}$ or $50\%$) means the event is equally likely to happen or not happen - like flipping a fair coin.

Q: How do I convert a fraction to a percentage?

A: Divide the numerator by the denominator, then multiply by 100. Example: $\frac{3}{8} = 3 \div 8 = 0.375 = 37.5\%$