#AP Statistics: Probability Distributions & Binomial Random Variables 🚀

Hey there, future AP Stats pro! Let's get you feeling confident and ready for the exam. This guide is your one-stop shop for probability distributions and binomial random variables. We'll break down the concepts, highlight key points, and get you prepped for anything the AP exam throws your way. Let's do this!

#Probability Distributions: The Big Picture

There are two main ways to build a probability distribution:

1. Theoretical Probability: Use the rules of probability to calculate the likelihood of each outcome. This is perfect for situations with clear-cut probabilities (like a fair coin). 🪙
2. Empirical Probability: Simulate the random event many times and use the observed frequencies to estimate probabilities. This is great for complex situations where theoretical probabilities are hard to calculate (like rolling a die many times). 🎲

#Binomial Random Variables: Success or Failure?

#Binomial Settings: The Rules of the Game

To be a binomial setting, you need to meet these conditions:

• Fixed Number of Trials (n): You're doing the same thing a set number of times (e.g., flipping a coin 10 times). 🪙
• Independent Trials: The outcome of one trial doesn't affect the outcome of any other trial (e.g., one coin flip doesn't change the next flip). 🔄
• Two Outcomes (Success or Failure): Each trial results in either a success or a failure (e.g., heads or tails). ✅/❌
• Constant Probability of Success (p): The probability of success is the same for each trial (e.g., the probability of getting heads is always 0.5 for a fair coin). 🎯

#Examples of Binomial Settings

• Coin Flips: Counting the number of heads in a series of coin flips.
• Medical Treatment: Counting the number of patients who have a positive outcome after a treatment.
• Manufacturing: Counting the number of defective products in a batch.
• Marketing: Counting the number of customers who make a purchase after seeing an ad.
• Surveys: Counting the number of customers who give a positive rating.

#Calculating Binomial Probabilities

The probability of getting exactly x successes in n trials is given by this formula:

$$P(X = x) = \binom{n}{x} * p^x * (1-p)^{(n-x)}$$

Where:

• $\binom{n}{x}$ is the binomial coefficient (also written as nCx), which represents the number of ways to choose x successes from n trials.
• p is the probability of success on a single trial.
• (1-p) is the probability of failure on a single trial.

#BinomPDF vs. BinomCDF

• binomPDF(n, p, x): Use this when you want the probability of exactly x successes. It's like asking, "What's the probability of getting exactly 3 heads?"
• binomCDF(n, p, x): Use this when you want the probability of x or fewer successes. It's like asking, "What's the probability of getting 3 or fewer heads?"

#Practice Problem: Snack Time! 🍔

Let's say you're a marketing manager and want to know the probability that exactly 3 out of 10 people like your new snack. Assume the probability of liking the snack is 0.5. ### Problem Setup

• n = 10 (10 people surveyed)
• p = 0.5 (probability of liking the snack)
• X = 3 (we want to know the probability of exactly 3 people liking the snack)

#Solution

Using the formula:

$$P(X=3) = \binom{10}{3} * (0.5)^3 * (0.5)^7$$

$$= 120 * (0.5)^3 * (0.5)^7$$

$$= 0.117$$

#Interpretation

The probability that exactly 3 out of 10 people like the snack is about 0.117. ## Final Exam Focus 🎯

• Key Concepts: Probability distributions, binomial settings (BINS), binomial random variables, binomial probabilities, binomPDF, binomCDF.
• High-Value Topics: Binomial distributions are a staple of the AP exam. Make sure you know how to identify a binomial setting and calculate probabilities.
• Common Question Types:
  • Identifying binomial settings.
  • Calculating binomial probabilities using formulas and calculator functions.
  • Interpreting probabilities in context.
• Time Management: Use your calculator efficiently! Don't waste time doing calculations by hand when you can use binomPDF and binomCDF.
• Common Pitfalls:
  • Forgetting to check the BINS conditions.
  • Confusing binomPDF and binomCDF.
  • Not interpreting results in context.

#

Practice Question

Practice Questions

#Multiple Choice Questions

1. A fair coin is flipped 5 times. What is the probability of getting exactly 3 heads? (a) 0.156 (b) 0.313 (c) 0.500 (d) 0.688 (e) 0.813

2. A basketball player makes 70% of their free throws. If they take 8 free throws, what is the probability that they make at least 6 of them? (a) 0.296 (b) 0.354 (c) 0.551 (d) 0.647 (e) 0.700

3. Which of the following is NOT a condition for a binomial setting? (a) There are a fixed number of trials. (b) The trials are independent. (c) There are more than two possible outcomes. (d) The probability of success is constant for each trial. (e) The random variable counts the number of successes.

#Free Response Question

A company manufactures light bulbs. It is known that 5% of the bulbs are defective. A random sample of 20 bulbs is selected for inspection.

(a) What is the probability that exactly 2 bulbs in the sample are defective?

(b) What is the probability that at least 2 bulbs in the sample are defective?

(c) What is the probability that more than 3 bulbs in the sample are defective?

(d) If the sample size was increased to 100, would the standard deviation of the number of defective bulbs increase, decrease, or stay the same? Explain.

#Scoring Rubric

(a) (2 points)

• 1 point for correctly identifying the binomial setting and parameters (n=20, p=0.05)
• 1 point for calculating the correct probability using binompdf(20, 0.05, 2) = 0.1887

(b) (2 points)

• 1 point for setting up the correct probability calculation: P(X ≥ 2) = 1 - P(X < 2)
• 1 point for calculating the correct probability using 1 - binomcdf(20, 0.05, 1) = 0.2642

(c) (2 points)

• 1 point for setting up the correct probability calculation: P(X > 3) = 1 - P(X ≤ 3)
• 1 point for calculating the correct probability using 1 - binomcdf(20, 0.05, 3) = 0.0071

(d) (2 points)

• 1 point for stating the standard deviation will increase
• 1 point for explaining that the standard deviation of a binomial distribution is given by $\sqrt{np(1-p)}$, and increasing n will increase the standard deviation.

Alright, you've got this! Review this guide, take a deep breath, and go ace that AP Statistics exam. You're more prepared than you think! 💪