Independent Events Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Independent Events.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Two events are independent if the occurrence of one does not change the probability of the other: P(A \cap B) = P(A) \cdot P(B).

They don't 'know about' each other. One happening tells you nothing about the other.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Independent events: multiply probabilities. Dependent events: need conditional probability.

Common stuck point: Independence is not the same as mutual exclusivity — independent events can both occur; mutually exclusive events cannot occur together.

Sense of Study hint: Ask yourself: does knowing the first result change the probability of the second? If yes, the events are dependent, not independent.

Worked Examples

Example 1

easy
A fair coin is flipped and a fair die is rolled. Find P(\text{Heads and a 4}).

Solution

  1. 1
    Identify the two events: A = \text{Heads}, B = \text{rolling a 4}
  2. 2
    Check independence: the coin flip does not affect the die roll, so A and B are independent
  3. 3
    Find individual probabilities: P(A) = \frac{1}{2}, P(B) = \frac{1}{6}
  4. 4
    Apply multiplication rule: P(A \cap B) = P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{1}{6} = \frac{1}{12}

Answer

P(\text{Heads and a 4}) = \frac{1}{12}
Two events are independent if the occurrence of one does not change the probability of the other. For independent events, P(A \cap B) = P(A) \cdot P(B). Physical separation of experiments is the clearest indicator of independence.

Example 2

medium
For two events A and B, P(A) = 0.4 and P(B) = 0.5. If A and B are independent, find (a) P(A \cap B) and (b) P(A \cup B).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A student randomly guesses on two multiple-choice questions, each with 4 options. Find the probability of getting both correct.

Example 2

hard
Events A and B satisfy P(A) = 0.6, P(B) = 0.7, and P(A \cap B) = 0.42. Are A and B independent? Justify algebraically.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

probability