#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.

#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?)

#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.
   Common Mistake

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 kth 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)

#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.

#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

Practice Question Text