Probability of Compound Events

Learn how to calculate the probability when two or more events happen together or in sequence.

Intermediate25 minLesson
Printable worksheet

Definition

A compound event consists of two or more simple events happening together or in sequence. There are two main types:
Independent Events: The outcome of one event does NOT affect the other.
P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
Dependent Events: The outcome of one event DOES affect the other.
P(A and B)=P(A)×P(B∣A)P(A \text{ and } B) = P(A) \times P(B|A)
For mutually exclusive events (events that cannot happen at the same time):
P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)

Try it now

You flip a coin and roll a die. What is the probability of getting heads AND rolling a 4?

Worked Examples

What is the probability of rolling a 6 on the first die AND a 6 on the second die?
1

Identify the events

Event A: First die shows 6. Event B: Second die shows 6. → Two separate events
2

Check if independent

The result of the first die does not affect the second die. → Independent events
3

Find individual probabilities

P(A)=16P(A) = \frac{1}{6} and P(B)=16P(B) = \frac{1}{6} → Each die has 6 faces
4

Apply the multiplication rule

P(A and B)=16×16=136P(A \text{ and } B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}

Common Mistakes

Adding probabilities when you should multiply

Why it's wrong: When finding the probability of A AND B happening, you must multiply. Adding is only for A OR B (mutually exclusive events).

Correct: AND = multiply: P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B). OR = add: P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)

Treating dependent events as independent

Why it's wrong: When one event affects another (like drawing without replacement), the probabilities change. Using the original probability gives wrong answers.

Correct: For dependent events, recalculate the probability after the first event. The total and favorable outcomes both change.

Forgetting to simplify fractions

Why it's wrong: The answer 2056\frac{20}{56} is correct but not fully simplified.

Correct: Always simplify: 2056=514\frac{20}{56} = \frac{5}{14}. Find the GCF and divide both numerator and denominator.

Interactive Visual

5 red, 3 blue: two draws without replacement

1st draw2nd draw5/84/73/73/85/72/7RedRedBlueBlueRedBlue(Red, Red)20/56(Red, Blue)15/56(Blue, Red)15/56(Blue, Blue)6/56

Multiply the choices at each step: 2 × 2 = 4 (4 outcomes)

All outcome probabilities add up to 1.

Tap an outcome on the right to follow its path.

Dice Roller

Click the button to run trials and see probability in action!

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
You flip a coin and roll a die. What is the probability of getting heads AND rolling a 4?

Why It Matters

Compound probability is essential for making informed decisions:
  • Games: What are the odds of rolling doubles with two dice?
  • Weather: What is the chance of rain both Saturday AND Sunday?
  • Medicine: If a test is 95% accurate, what happens with two tests?
  • Security: How secure is a 4-digit PIN code against random guessing?
Understanding compound events helps you evaluate risks and make smarter choices!

Real World Applications

Password Security

Banks and websites use compound probability to measure password strength.

Example:

A 4-digit PIN has 10 choices per digit. The probability of guessing it randomly is 110×110×110×110=110000\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} = \frac{1}{10000}.

1Try It Yourself

A password requires 2 letters (A-Z) followed by 2 digits (0-9).

How many possible combinations are there?

Step 1: Write the mathematical expression

Calculate: 26×26×10×1026 \times 26 \times 10 \times 10

Medical Testing

Doctors use compound probability to evaluate test accuracy.

Example:

If a medical test is 98% accurate, the probability of two independent tests both being wrong is 0.02×0.02=0.00040.02 \times 0.02 = 0.0004 or 0.04%.

2Try It Yourself

A drug test is 95% accurate. A person takes the test twice.

What is the probability that both tests give the correct result?

Step 1: Write the mathematical expression

Calculate: 0.95×0.950.95 \times 0.95

Key Takeaways

  • 1Compound events involve two or more simple events happening together
  • 2For independent events (no effect on each other): P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
  • 3For dependent events: recalculate probabilities after each event
  • 4For mutually exclusive events (cannot happen together): P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)
  • 5AND means multiply, OR means add (for mutually exclusive events)

Frequently Asked Questions

Ask yourself: Does the first event change the situation for the second event? If drawing a card and NOT replacing it, the events are dependent because the deck changes. If flipping a coin twice, the events are independent because each flip has no effect on the next.
Ask yourself: Does the first event change the situation for the second event? If drawing a card and NOT replacing it, the events are dependent because the deck changes. If flipping a coin twice, the events are independent because each flip has no effect on the next.
Think of AND as narrowing down possibilities - both conditions must be met, so the probability gets smaller (multiplication makes it smaller). OR expands possibilities - either condition works, so we combine the chances (addition makes it bigger).
P(B|A) means 'the probability of B given that A has already happened.' It is the conditional probability - how likely is B after A occurs? For example, P(second red | first red) is the probability of drawing a second red marble after already drawing a red marble.

Glossary

Compound event
An event made up of two or more simple events
Independent events
Events where the outcome of one does not affect the probability of the other
Dependent events
Events where the outcome of one affects the probability of the other
Mutually exclusive
Events that cannot happen at the same time
Conditional probability
The probability of an event given that another event has occurred, written as P(B∣A)P(B|A)
Sample space
The set of all possible outcomes of an experiment

Formula Card

Independent Events (AND)

P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)

Multiply probabilities when events do not affect each other

Dependent Events (AND)

P(A and B)=P(A)×P(B∣A)P(A \text{ and } B) = P(A) \times P(B|A)

Use conditional probability when one event affects the other

Mutually Exclusive (OR)

P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)

Add probabilities when events cannot happen together

General Addition Rule

P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Use when events can overlap (not mutually exclusive)

Today's challengeOne problem, three tries

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Independent Events

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Dependent Events

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