Topic 4.12 Notes – The Geometric Distribution

1. What a Geometric Random Variable Is

A geometric random variable $X$ counts the trial number of the first success in a sequence of independent trials.

This setup always has four ingredients:

• Two outcomes each trial: success or failure
• Constant probability of success $p$
• Independent trials
• $X = 1, 2, 3, \dots$

If the first success happens on trial $x$, then:

• Trials $1$ through $x-1$ were failures
• Trial $x$ was a success

That structure is the whole distribution.

Geometric vs. Binomial

Students mix these up constantly.

If the question says “how many trials until the first…”, your brain should jump to geometric immediately.

2. The Geometric Probability Formula

For $X \sim \text{Geometric}(p)$:

$$P(X = x) = (1 - p)^{x - 1} p$$

Why it works:

• $(1-p)^{x-1}$ → first $x-1$ trials are failures
• $p$ → the $x$th trial is success

Example: Suppose a free throw shooter makes shots with probability $p = 0.7$. What’s the probability the first made shot is on the 4th attempt?

$$P(X=4) = (0.3)^3(0.7)$$

$$= 0.027(0.7) = 0.0189$$

Interpretation (this wording matters on FRQs):

The probability that the player makes their first free throw on the 4th attempt is 0.0189.

Cumulative Probabilities

A huge shortcut you should know:

$$P(X > k) = (1 - p)^k$$

“More than $k$ trials” means the first $k$ were all failures.

Then:

$$P(X \le k) = 1 - (1 - p)^k$$

On your calculator:

• geometpdf(p, x) gives $P(X=x)$
• geometcdf(p, x) gives $P(X \le x)$

Common slip: forgetting that $x$ starts at 1. There is no $X=0$.

3. Mean and Standard Deviation

For $X \sim \text{Geometric}(p)$:

$$\mu_X = \frac{1}{p}$$

$$\sigma_X = \frac{\sqrt{1-p}}{p}$$

Interpreting the Mean

The mean $1/p$ is the long-run average number of trials needed to get the first success.

If $p = 0.2$:

$$\mu = \frac{1}{0.2} = 5$$

Interpretation:

On average, it takes 5 trials to get the first success.

Units are trials, not probability.

Notice the relationship:

• Larger $p$ → smaller mean
• Smaller $p$ → larger mean

That makes sense. Rare successes take longer to happen.

Shape

Every geometric distribution is right-skewed.

Here is what a geometric distribution looks like when $p = 0.4$.

Geometric distribution with p = 0.4

The highest probability is always at $X=1$, and probabilities decrease as $x$ increases.

4. How to Recognize and Use It on the Exam

When you see a scenario, mentally check:

• Independent repeated trials?
• Two outcomes?
• Constant $p$?
• Counting until the first success?

If yes, use geometric.

Memoryless Property

Geometric distributions are memoryless:

$$P(X > s+t \mid X > s) = P(X > t)$$

If you haven’t had success yet, nothing changes going forward. The probability of success on the next trial is still $p$.

Students often fall into the “it’s due now” thinking. It isn’t. Each trial is independent.

Interpreting in Context

On quizzes and the AP exam, vague answers lose points.

Good probability interpretation:

The probability that the first defective item is produced on the 6th item is 0.064.

Good mean interpretation:

On average, it takes 8 items to get the first defective item.

Name the variable. Name the situation. Use correct units.