• Number: The number of trials (n) must be fixed in advance.
• Same Probability: The probability of success (p) must be the same for each trial.

Example of when NOT to use a binomial distribution:

Imagine a biased coin where the probability of heads changes with each flip (0.8 on the first flip, 0.6 on the second, etc.). This violates the "Same Probability" condition, so we can't use a binomial distribution here.

#Mean and Standard Deviation of Binomial Variables

Once you've confirmed that you have a binomial setting, you can calculate the mean and standard deviation using these formulas:

• Mean (Expected Value): $E(X) = n * p$
  • This tells you the average number of successes you'd expect over many repetitions of the experiment.
• Standard Deviation: $σ_x = \sqrt{n * p * (1-p)}$
  • This tells you how much the number of successes typically varies from the mean.

#Source: College Board (AP Statistics Formula Sheet and Tables)

#Binomial Distributions in Statistical Sampling

When we're sampling from a population, there's one more rule to consider: the 10% condition. This condition helps us ensure our trials are approximately independent.

• 10% Condition: If you're sampling without replacement, the sample size (n) should be less than 10% of the population size (N). That is, n < 0.10N. 🛞

#Final Exam Focus

Alright, let's focus on what's most important for the exam:

• High-Value Topics:
  • Checking the BINS conditions is crucial for any binomial problem.
  • Calculating the mean and standard deviation of binomial random variables.
  • Understanding when and why to use the 10% condition.
• Common Question Types:
  • Multiple-choice questions that test your understanding of the BINS conditions.
  • Free-response questions that require you to justify the use of a binomial model and calculate probabilities.
• Time Management:
  • Quickly check the BINS conditions. If one is violated, you don't need to proceed with binomial calculations.
  • Be mindful of the wording of the question. Does it require you to calculate a probability, mean, or standard deviation?
• Common Pitfalls:
  • Not checking all four BINS conditions.
  • Forgetting to check the 10% condition when sampling without replacement.
  • Using binomial formulas when the conditions are not met.

#Practice Questions

Okay, let's put your knowledge to the test! Here are some practice questions to get you ready:

Practice Question

#Multiple Choice Questions

1. A fair six-sided die is rolled 10 times. Let X be the number of times a 4 or 5 is rolled. Which of the following statements is true?

   (A) X has a binomial distribution with n = 10 and p = 1/6. (B) X has a binomial distribution with n = 10 and p = 1/3. (C) X has a binomial distribution with n = 6 and p = 1/3. (D) X does not have a binomial distribution because the trials are not independent.

   (E) X does not have a binomial distribution because there are more than two outcomes.

2. A survey indicates that 60% of adults in a large city support a new sports stadium. If a random sample of 20 adults is selected, what is the standard deviation of the number of people in the sample who support the new stadium?

   (A) 2.4

   (B) 4.8

   (C) 4.9

   (D) 12

   (E) 12.8

#Free Response Question

A large company has 5000 employees. The company's HR department is conducting a survey to gather feedback on employee satisfaction. They randomly select 200 employees to participate in the survey. It is known that 70% of all employees are satisfied with their jobs.

(a) Verify that the conditions for a binomial distribution are met for the number of satisfied employees in the sample.

(b) Calculate the mean and standard deviation of the number of satisfied employees in the sample.

(c) What is the probability that exactly 150 employees in the sample are satisfied with their jobs? (You don't need to calculate the exact value, but set up the calculation).

#Scoring Breakdown for FRQ

(a) Verify the conditions for a binomial distribution:

• Binary: 1 point for stating that each employee either is satisfied or not satisfied.
• Independent: 1 point for stating that the employees are randomly sampled and that the sample size is less than 10% of the population size (200 < 0.10 * 5000 = 500).
• Number: 1 point for stating that the number of trials (200 employees) is fixed.
• Same Probability: 1 point for stating that the probability of success (0.70) is the same for each employee.

(b) Calculate the mean and standard deviation:

• Mean: 1 point for calculating the mean as n * p = 200 * 0.70 = 140. * Standard Deviation: 1 point for correctly using the formula and calculating the standard deviation as sqrt(n * p * (1-p)) = sqrt(200 * 0.70 * 0.30) ≈ 6.48. (c) Probability of exactly 150 satisfied employees:

• 1 point for setting up the calculation correctly: P(X = 150) = (200 choose 150) * (0.70)^150 * (0.30)^50

#Wrapping Up

You've got this! Remember to stay calm, focus on the BINS conditions, and apply the formulas correctly. You're well-prepared, and now it's time to show the AP exam what you've got. Good luck, and go ace that test! 🌟