Lesson 91: Probability Distributions (Binomial, Normal)
A probability distribution shows how probabilities are assigned to outcomes of a random process. In this lesson, we focus on two important distributions: binomial (discrete, repeated trials) and normal (continuous, bell-shaped).
Binomial Distribution
The binomial distribution models the probability of getting exactly \(k\) successes in \(n\) independent trials, each with probability \(p\) of success.
\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)
- \(n\) = number of trials
- \(k\) = number of successes
- \(p\) = probability of success
Example: A coin is flipped 5 times. What is the probability of exactly 2 heads? \( P(X=2) = \binom{5}{2} (0.5)^2 (0.5)^3 = 10(0.25)(0.125) = 0.3125 \).
Normal Distribution
The normal distribution is a continuous distribution shaped like a bell curve, centered at the mean \(\mu\) with spread measured by standard deviation \(\sigma\).
\( f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} } \)
- Mean (\(\mu\)) = center of the distribution
- Standard deviation (\(\sigma\)) = controls the spread
- 68–95–99.7 Rule: About 68% of data within 1σ, 95% within 2σ, 99.7% within 3σ.
Example: Heights of adult men are approximately normal with mean 70 in and standard deviation 3 in. About 95% of men are between 64 in and 76 in.
More Examples
Binomial Example — Basketball Free Throws:
A player makes 80% of free throws. If they take 10 shots, what is the probability they make exactly 8? \( P(X=8) = \binom{10}{8} (0.8)^8 (0.2)^2 = 45(0.1678)(0.04) ≈ 0.302 \).
Normal Example — Test Scores:
SAT scores are normally distributed with mean 1050 and σ = 100. What percentage of students score above 1250? Standardize: \( z = (1250 - 1050)/100 = 2 \). From z-tables, \( P(Z > 2) ≈ 0.0228 \). So about 2.3%.
Practice Problems
- Flip a coin 6 times. Find the probability of getting exactly 3 heads.
- A die is rolled 12 times. What is the probability of rolling exactly 2 sixes?
- In a binomial distribution with \(n = 20, p = 0.4\), find \(P(X=8)\).
- Heights of women are normally distributed with mean 64 in and σ = 2.5 in. What proportion are taller than 67 in?
- Test scores have mean 80 and σ = 10. Find the probability a student scores between 70 and 90.