MODULE 0810 QUESTIONS

Binomial Distribution

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Binomial Distribution

When to Use the Binomial

The binomial distribution models the number of successes in a fixed number of Bernoulli trials — independent experiments that can only end in success or failure.

Four conditions must ALL hold:

1. Fixed n — you know how many trials before you start

2. Independence — each trial's outcome doesn't affect others

3. Constant p — same probability of success on every trial

4. Binary — only two possible outcomes per trial

Classic examples: Number of heads in 10 coin flips. Number of patients who recover out of 20 given a treatment. Number of correct guesses on a 10-question T/F test.

The PMF Formula

For X ~ Binomial(n, p), the probability of exactly k successes:

P(X = k) = C(n,k) p^k (1-p)^(n-k)

The C(n,k) term counts the number of ways to arrange k successes among n trials. The p^k and (1-p)^(n-k) give the probability of any specific arrangement.

Example: P(exactly 3 heads in 5 flips) = C(5,3) 0.5^3 0.5^2 = 10 0.125 0.25 = 0.3125.

R Functions

In R, you'll never compute the PMF formula by hand — use these functions:

R
1# P(X = 3) when X ~ Binomial(10, 0.4)
2dbinom(3, size = 10, prob = 0.4)
3
4# P(X <= 5) — cumulative probability
5pbinom(5, size = 10, prob = 0.4)
6
7# P(X >= 6) — upper tail: 1 - P(X <= 5)
81 - pbinom(5, size = 10, prob = 0.4)
9
10# P(X >= 6) directly with lower.tail = FALSE
11pbinom(5, size = 10, prob = 0.4, lower.tail = FALSE)

dbinom vs pbinom: d = density (exact probability at one value). p = probability (cumulative, all values up to and including k). For "at least", remember that P(X >= k) = 1 - P(X <= k-1), not 1 - P(X <= k).

Mean and Variance

For X ~ Binomial(n, p):

  • ›Mean: E[X] = np
  • ›Variance: Var(X) = np(1-p)
  • ›Standard deviation: SD(X) = sqrt(np(1-p))
R
1n <- 20
2p <- 0.3
3mean_X <- n * p
4var_X <- n * p * (1 - p)
5sd_X <- sqrt(var_X)

Intuition for the mean: If you flip a coin 20 times (p = 0.5), you expect 10 heads. If p = 0.3, you expect 20 * 0.3 = 6 successes. The mean is just n times the probability of a single success.

Visualizing the Binomial

R
1tibble(k = 0:15) %>%
2 mutate(prob = dbinom(k, size = 15, prob = 0.4)) %>%
3 ggplot(aes(x = k, y = prob)) +
4 geom_col(fill = "p0
5 labs(title = "Binomial(15, 0.4)", x = "Number of successes k", y = "P(X = k)")

The distribution is symmetric only when p = 0.5. For p < 0.5 it's right-skewed; for p > 0.5 it's left-skewed.

Combinations and Factorials

How many ways can you arrange k successes in n trials?

C(n,k) = n! / (k! * (n-k)!)

In R:

R
1choose(5, 3) # C(5,3) = 10 ways to choose 3 items from 5
2factorial(5) # 5! = 120

The Four Binomial Functions

dbinom(k, n, p) — exact probability P(X = k):

R
1dbinom(5, size = 10, prob = 0.4) # P(X = 5) for Binom(10, 0.4)

pbinom(k, n, p) — cumulative probability P(X <= k):

R
1pbinom(5, size = 10, prob = 0.4) # P(X <= 5)

qbinom(q, n, p) — quantile (inverse CDF). Find the smallest x where P(X <= x) >= q:

R
1qbinom(0.7, size = 10, prob = 0.5) # returns 6, since P(X <= 5)=0.623 < 0.7 but P(X <= 6)=0.828 >= 0.7

rbinom(n_samples, size, prob) — simulate random samples:

R
1rbinom(5, size = 10, prob = 0.5) # generate 5 random Binomial(10, 0.5) values

Quick reference: d = density (exact), p = probability (cumulative), q = quantile, r = random sample.

BINS Mnemonic for Binomial

The four required conditions for Binomial(n, p):

  • ›Binary — each trial has exactly two outcomes (success/failure, yes/no)
  • ›Independent — outcome of one trial doesn't affect others
  • ›Number fixed — n is determined before the experiment
  • ›Same p — probability of success is identical for every trial

Common violation: Sampling without replacement from a small population violates Independence. Rule of thumb: If population size >= 20n, independence is approximately satisfied.

P(X >= k) vs P(X > k) — Critical Distinction

For a DISCRETE distribution these are NOT the same:

  • ›P(X >= k) = "at least k" = 1 - P(X <= k-1) → use 1 - pbinom(k-1, n, p)
  • ›P(X > k) = "more than k" = 1 - P(X <= k) → use 1 - pbinom(k, n, p)

Example: X ~ Binomial(10, 0.4)

R
1# P(X >= 4) — at least 4 successes
21 - pbinom(3, 10, 0.4)
R
1# P(X > 4) — more than 4 (i.e., at least 5)
21 - pbinom(4, 10, 0.4)

This is the #1 source of exam errors. "At least 4" means >= 4, so subtract P(X <= 3), NOT P(X <= 4).

Simulation with rbinom()

You can simulate binomial experiments using rbinom():

R
1# Simulate 100,000 experiments: n=10 trials, p=0.4
2sims <- rbinom(100000, size = 10, prob = 0.4)
3mean(sims) # should be close to n*p = 4

The simulated mean (approximately 4) confirms our formula E[X] = np = 10 0.4 = 4. Simulation is a powerful way to verify theoretical results.