Calculating Simple Probability

Learn to calculate the probability of events using the basic probability formula.

Intermediate25 minLesson
Printable worksheet

Definition

Probability measures how likely an event is to happen. We calculate it using this formula:
P(event)=Number of favorable outcomesTotal number of possible outcomesP(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
Probability is always a number between 00 and 11:
  • P=0P = 0 means the event is impossible
  • P=1P = 1 means the event is certain
  • P=0.5P = 0.5 or 12\frac{1}{2} means equally likely to happen or not
We can also express probability as a fraction, decimal, or percentage:
14=0.25=25%\frac{1}{4} = 0.25 = 25\%

Try it now

A bag contains 3 red marbles and 7 blue marbles. What is the probability of drawing a red marble?

Worked Examples

What is the probability of rolling a 44 on a standard six-sided die?
1

Identify favorable outcomes

There is only one face showing 44 → Favorable outcomes = 11
2

Count total possible outcomes

A die has 6 faces: 1,2,3,4,5,61, 2, 3, 4, 5, 6 → Total outcomes = 66
3

Apply the formula

P(4)=favorabletotal=16P(4) = \frac{\text{favorable}}{\text{total}} = \frac{1}{6} → P(4)=16P(4) = \frac{1}{6}
4

Convert to percentage (optional)

16≈0.167≈16.7%\frac{1}{6} \approx 0.167 \approx 16.7\% → About 16.7%16.7\% chance

Common Mistakes

Confusing favorable outcomes with total outcomes

Why it's wrong: 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=what you wantall possibilitiesP = \frac{\text{what you want}}{\text{all possibilities}}

Forgetting to simplify fractions

Why it's wrong: While 2652\frac{26}{52} is correct, the simplified form 12\frac{1}{2} is clearer and easier to interpret.

Correct: Always simplify probability fractions when possible. 2652=12\frac{26}{52} = \frac{1}{2}

Thinking probability can be greater than 1

Why it's wrong: Some students calculate probabilities like 85\frac{8}{5} when miscounting outcomes.

Correct: Probability is ALWAYS between 00 and 11 (or 0%0\% to 100%100\%). If your answer is greater than 11, recheck your counts!

Not counting all favorable outcomes

Why it's wrong: 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...5, 6, 7, 8...; 'greater than 5' means 6,7,8...6, 7, 8...

Interactive Visual

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Practice Problems

15 problems
Problem 1 of 15
Easy
A bag contains 3 red marbles and 7 blue marbles. What is the probability of drawing a red marble?

Why It Matters

Probability helps us make smart decisions in uncertain situations:
  • Weather: A 70% chance of rain helps you decide to bring an umbrella
  • Games: Knowing odds helps board game and card game strategy
  • Medicine: Doctors use probability to assess treatment success rates
  • Sports: Coaches analyze win probabilities to plan strategies
  • Business: Companies use probability to predict sales and manage inventory
Understanding probability helps you think critically about risks and make better choices!

Real World Applications

Weather Forecasting

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

Example:

If rain occurred on 7 out of 20 similar weather days, the probability is 720=35%\frac{7}{20} = 35\%.

1Try It Yourself

A weather station recorded rain on 12 out of 30 days with similar conditions.

What is the probability of rain tomorrow?

Step 1: Write the mathematical expression

Calculate: favorable (rainy days) / total (similar days)

Game Show Prizes

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

Example:

If a wheel has 20 sections with 3 showing 'Grand Prize', P(grand prize)=320=15%P(\text{grand prize}) = \frac{3}{20} = 15\%.

2Try It Yourself

A game show wheel has 25 sections. 4 show 'Win 1000 euros', 8 show 'Win 100 euros', and the rest show 'Try Again'.

What is the probability of winning any prize?

Step 1: Write the mathematical expression

Prize sections: 4+8=124 + 8 = 12. Calculate probability:

Quality Control

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

Example:

If 3 out of 500 items are defective, P(defect)=3500=0.6%P(\text{defect}) = \frac{3}{500} = 0.6\%.

3Try It Yourself

A factory tests 200 smartphones. 8 have defects.

What is the probability that a randomly selected phone is defective?

Step 1: Write the mathematical expression

Calculate: defective / total tested

Key Takeaways

  • 1Probability measures how likely an event is to happen
  • 2Formula: P(event)=favorable outcomestotal outcomesP(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}
  • 3Probability is always between 00 (impossible) and 11 (certain)
  • 4Probability can be written as a fraction, decimal, or percentage
  • 5Always simplify fractions and double-check your outcome counts

Frequently Asked Questions

The number of favorable outcomes can never exceed the total outcomes. At most, every outcome is favorable, giving P=nn=1P = \frac{n}{n} = 1 (certainty).
The number of favorable outcomes can never exceed the total outcomes. At most, every outcome is favorable, giving P=nn=1P = \frac{n}{n} = 1 (certainty).
A probability of 0.50.5 (or 12\frac{1}{2} or 50%50\%) means the event is equally likely to happen or not happen - like flipping a fair coin.
Divide the numerator by the denominator, then multiply by 100. Example: 38=3÷8=0.375=37.5%\frac{3}{8} = 3 \div 8 = 0.375 = 37.5\%

Glossary

Probability
A number from 00 to 11 measuring how likely an event is to occur
Favorable outcome
An outcome that matches what we're looking for
Total outcomes
All possible outcomes that could occur
Sample space
The set of all possible outcomes
Event
A specific outcome or set of outcomes we're interested in

Formula Card

Probability Formula

P(event)=favorable outcomestotal outcomesP(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

$P$ is the probability (between 0 and 1). Favorable outcomes are those matching our event. Total outcomes are all possibilities.

Today's challengeOne problem, three tries

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