AP Precalculus: Binomial & Normal Distributions
Master binomial probability, normal curves, z-scores, and the central limit theorem
📚 Two Key Distributions
The binomial distribution models the number of successes in a fixed number of independent trials. The normal distribution is a continuous bell-shaped curve that appears frequently in nature and statistics. Understanding both is essential for probability calculations on the AP exam.
1 Binomial Distribution
A binomial distribution models the number of successes in n independent trials, where each trial has the same probability p of success.
Conditions for Binomial
- Fixed number of trials (n)
- Each trial is independent
- Only two outcomes: success or failure
- Same probability p for each trial
Problem: A coin is flipped 5 times. Find P(exactly 3 heads).
Identify: n = 5, k = 3, p = 0.5
Calculate: \(P(X=3) = \binom{5}{3}(0.5)^3(0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125\)
2 Binomial Mean, Variance & Standard Deviation
For a binomial distribution with n trials and probability p, these formulas give the expected outcomes and spread.
Problem: A test has 20 multiple choice questions with 4 choices each. If guessing randomly, find the mean and standard deviation of correct answers.
Given: n = 20, p = 0.25
Mean: \(\mu = 20(0.25) = 5\) correct answers expected
Std Dev: \(\sigma = \sqrt{20(0.25)(0.75)} = \sqrt{3.75} \approx 1.94\)
3 Normal Distribution
The normal distribution is a continuous, symmetric, bell-shaped curve defined by its mean μ and standard deviation σ.
Properties of Normal Distribution
- Symmetric about the mean
- Mean = Median = Mode
- Total area under curve = 1
- 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ (Empirical Rule)
68-95-99.7 Rule
68% within μ ± σ
95% within μ ± 2σ
99.7% within μ ± 3σ
Tails
Extends infinitely in both directions, approaching but never touching the x-axis
4 Z-Scores & Finding Probabilities
A z-score tells how many standard deviations a value is from the mean. It converts any normal distribution to the standard normal.
Find Probability from X
1. Calculate z-score
2. Use z-table or calculator
3. Read probability
Find X from Probability
1. Find z from table/calculator
2. Use: \(x = \mu + z\sigma\)
Problem: Test scores are normal with μ = 75, σ = 8. Find P(score > 85).
Z-score: \(z = \frac{85 - 75}{8} = 1.25\)
From table: P(Z < 1.25)=0.8944
Answer: P(score > 85) = 1 - 0.8944 = 0.1056 or 10.56%
z > 0 → above mean | z < 0 → below mean | z=0 → at the mean
5 Distribution of Sample Means
When taking many samples of size n from a population, the sample means form their own distribution called the sampling distribution.
Population: μ = 100, σ = 15. Sample size n = 25.
Mean of sample means: \(\mu_{\bar{x}} = 100\)
Standard error: \(\sigma_{\bar{x}} = \frac{15}{\sqrt{25}} = \frac{15}{5} = 3\)
6 Central Limit Theorem (CLT)
The Central Limit Theorem states that for large sample sizes (n ≥ 30), the distribution of sample means is approximately normal, regardless of the original population's shape.
What CLT Tells Us
Sample means follow a normal distribution when n is large enough, even if the population isn't normal
Parameters
Mean: μ
Standard Deviation: \(\frac{\sigma}{\sqrt{n}}\)
Generally, n ≥ 30 is considered "large enough" for CLT to apply. If the population is already normal, CLT works for any sample size.
7 Normal Approximation to Binomial
When a binomial distribution has large enough n, it can be approximated by a normal distribution for easier calculations.
Conditions for Normal Approximation
Approximation Parameters
Since we're approximating discrete with continuous:
P(X ≤ k) → P(X < k + 0.5)
P(X ≥ k) → P(X > k - 0.5)
P(X = k) → P(k - 0.5 < X < k + 0.5)
Problem: n = 100, p = 0.4. Find P(X ≤ 35) using normal approximation.
Check: np = 40 ≥ 10 ✓, n(1-p) = 60 ≥ 10 ✓
Parameters: μ = 40, σ = √24 ≈ 4.9
With continuity: P(X ≤ 35.5)
Z-score: z = (35.5 - 40)/4.9 ≈ -0.92
Answer: P(Z < -0.92) ≈ 0.1788
📋 Quick Reference
Binomial P(X=k)
\(\binom{n}{k}p^k(1-p)^{n-k}\)
Binomial Mean
\(\mu = np\)
Binomial Std Dev
\(\sigma = \sqrt{np(1-p)}\)
Z-Score
\(z = \frac{x - \mu}{\sigma}\)
Standard Error
\(\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}\)
Normal Approx Condition
\(np ≥ 10\) and \(n(1-p) ≥ 10\)