# AP Statistics: Geometric Distributions - Your Last-Minute Guide 🚀 Hey there, future AP Stats pro! Let's get you feeling confident about geometric distributions. This guide is designed to be your go-to resource the night before the exam. Let's make sure you're ready to ace it! 💪 ## What are Geometric Random Variables? A **geometric random variable** counts the number of trials needed to get the *first* success. Think of it like this: you keep going until you finally win! Each trial is independent, with only two outcomes: success (probability *p*) or failure (probability 1-*p*). ### Key Characteristics: * **Discrete:** The variable can only take on whole number values (1, 2, 3, ...). * **Independent Trials:** Each trial doesn't affect the next. * **Two Outcomes:** Success or failure on each trial. * **First Success Focus:** We're interested in the trial number where the first success occurs. Remember: Geometric distributions are all about the *first* success. This is the key difference between geometric and binomial distributions. ### Example: Flipping a coin until you get heads. The number of flips it takes is a geometric random variable. ## Geometric vs. Binomial: The Ultimate Showdown! It's easy to mix these two up, but here's the lowdown: * **Binomial:** Counts the number of successes in a *fixed* number of trials. (e.g., How many heads in 10 coin flips?) * **Geometric:** Counts the number of trials needed to get the *first* success. (e.g., How many flips until the first head?) **BINS** for Binomial: **B**inary, **I**ndependent, **N**umber of trials fixed, **S**uccess probability fixed. **GIFT** for Geometric: **G**o until, **I**ndependent, **F**irst success, **T**rials not fixed ### Examples to Nail the Difference: 1. **Binomial:** Rolling a die 20 times and counting how many times you get a 6. 2. **Geometric:** Rolling a die until you get a 6. Don't confuse these! If the number of trials is fixed, it's binomial. If you're waiting for the first success, it's geometric. ## Calculating Probabilities: The Formulas You Need ### Probability Mass Function (PMF): The probability of the first success occurring on the *k*th trial is: $$P(Y = k) = (1-p)^{k-1} * p$$ Where: * *Y* is the geometric random variable * *k* is the number of trials * *p* is the probability of success on each trial ### Example: If *p* = 0.2, the probability that the first success occurs on the 3rd trial is: $P(Y = 3) = (1-0.2)^{3-1} * 0.2 = 0.8^2 * 0.2 = 0.128$ ### Using Technology 📱 * **geometricPDF(p, k):** Probability of the first success on trial *k*. (Same as the PMF) * **geometricCDF(p, k):** Probability of the first success on or before trial *k*. (Cumulative probability) Know when to use PDF (exact value) vs. CDF (cumulative value). CDF is your friend when you need "less than or equal to" probabilities. ![](https://zupay.blob.core.windows.net/resources/files/0baca4f69800419293b4c75aa2870acd_48e3d1_2043.png?alt=media&token=f77e2d42-4f65-410c-97a2-e08170c72740) ## Shape, Center, and Variability: Understanding the Distribution ### Shape: * Geometric distributions are always **skewed right**. This is because the probability of success decreases with each trial. The longer you wait for the first success, the less likely it becomes. ➡️ ### Center: * **Mean (Expected Value):** $E(Y) = \frac{1}{p}$ . This is the average number of trials you'd expect to need for the first success. ### Variability: * **Standard Deviation:** $σ_Y = \sqrt{\frac{1-p}{p^2}}$ . This tells you how much the number of trials typically varies from the mean. Remember: The mean (expected value) is simply 1/p. This is a crucial formula to memorize! ![](https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-rNm3bufE9KBs.webp?alt=media&token=1e041f30-50db-4714-bd1e-69c3d6c8e605) ## Practice Problem: Let's Put It All Together! **Problem:** A company produces items with a 8% defect rate. What's the probability that the first defective item is the 15th one produced? **Solution:** 1. **Identify:** This is a geometric distribution because we're looking for the first success (defective item). 2. **Parameters:** *p* = 0.08, *k* = 15 3. **Formula:** $P(Y = 15) = (1-0.08)^{15-1} * 0.08$ 4. **Calculate:** $P(Y = 15) = (0.92)^{14} * 0.08 ≈ 0.025$ 5. **Interpret:** There's about a 2.5% chance the 15th item is the first defective one. Practice Question Text

The Geometric Distribution

Ava Garcia

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