Geometric Distribution Examples in Math

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

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

Concept Recap

The probability distribution for the number of independent Bernoulli trials needed to get the first success, where each trial has success probability p.

How many times do you have to roll a die before you get a 6? The geometric distribution answers this kind of question. Each trial is independent, and you keep going until you succeed. Most of the time it doesn't take too long, but occasionally you have an unlucky streak—that's why the distribution has a long right tail.

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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: The geometric distribution is memoryless: no matter how many failures so far, the probability of success on the next trial is still p. The expected number of trials until first success is \frac{1}{p}.

Common stuck point: Students confuse geometric (trials until FIRST success) with binomial (number of successes in FIXED trials). Also watch out: some textbooks define X as the number of failures before the first success, shifting the formula.

Worked Examples

Example 1

medium
A basketball player makes free throws with probability p=0.7. Find the probability they make their first free throw on exactly the 3rd attempt.

Solution

  1. 1
    Geometric distribution: P(X=k) = (1-p)^{k-1} \cdot p
  2. 2
    Here: first make on attempt k=3; p=0.7; q=1-p=0.3
  3. 3
    P(X=3) = (0.3)^{3-1} \times 0.7 = (0.3)^2 \times 0.7 = 0.09 \times 0.7 = 0.063
  4. 4
    Interpretation: 6.3% chance the first success comes on the 3rd attempt (2 misses then a make)

Answer

P(X=3) = (0.3)^2(0.7) = 0.063. 6.3% chance first make is on attempt 3.
The geometric distribution models the number of trials until the first success. The formula requires k-1 failures (each with probability q=1-p) followed by one success (probability p). It assumes independence between trials.

Example 2

hard
For a geometric distribution with p=0.4: (a) find P(X \leq 3), (b) find the expected number of trials until first success.

Practice Problems

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

Example 1

easy
A fair coin is flipped until heads appears. What is P(\text{first heads on flip 4}), and what is the expected number of flips?

Example 2

hard
A quality inspector samples items until finding the first defective. Defect probability is p=0.05. Find P(X > 10) (probability of needing more than 10 inspections) and the expected number to inspect.
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Background Knowledge

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

binomial distributionindependent eventsexpected value