Geometric Distribution Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

medium
A basketball player makes free throws with probability p=0.7p=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⋅pP(X=k) = (1-p)^{k-1} \cdot p
  2. 2
    Here: first make on attempt k=3k=3; p=0.7p=0.7; q=1−p=0.3q=1-p=0.3
  3. 3
    P(X=3)=(0.3)3−1×0.7=(0.3)2×0.7=0.09×0.7=0.063P(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)=0.063P(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−1k-1 failures (each with probability q=1−pq=1-p) followed by one success (probability pp). It assumes independence between trials.

About Geometric Distribution

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

Learn more about Geometric Distribution →

More Geometric Distribution Examples

Example 2 hard

For a geometric distribution with [formula]: (a) find [formula], (b) find the expected number of tri

Example 3 easy

A fair coin is flipped until heads appears. What is [formula], and what is the expected number of fl

Example 4 hard

A quality inspector samples items until finding the first defective. Defect probability is [formula]

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