pbinom() — Cumulative binomial

The chance of getting q wins or fewer.

pbinom(q, size, prob, lower.tail = TRUE)

Key idea

• pbinom gives the cumulative chance of AT MOST q successes, written P(X <= q) — q wins or fewer. (e.g. pbinom(2, 10, 0.5) = 0.055)
• It adds up dbinom for every value from 0 through q. (e.g. P(0)+P(1)+P(2))
• Use it for questions like q or fewer heads in n flips. (e.g. at most 2 of 10)
• For more than q, take 1 minus the cumulative value. (e.g. 1 - pbinom(2,10,0.5))
• At q = n the total is always 1, since every outcome is covered. (e.g. pbinom(10,10,0.5) = 1)

Definition

$$F(q)=\sum _{k=0}^q\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$

$q$

successes counted up to

$n$

number of trials (size)

$p$

chance of success each trial

$$P(X\le 2)=\frac{\left(\genfrac{}{}{0ex}{}{10}{0}\right)+\left(\genfrac{}{}{0ex}{}{10}{1}\right)+\left(\genfrac{}{}{0ex}{}{10}{2}\right)}{2^{10}}=\frac{56}{1024}=0.0547$$

Where you'd use it

• Flip a fair coin 10 times: the chance of 3 or fewer heads is pbinom(3, 10, 0.5) ≈ 17%, telling you a result that low is uncommon but not rare.
• A box of 20 eggs each has a 5% chance of being cracked. The chance of no more than 2 cracked is pbinom(2, 20, 0.05) ≈ 92%, so most boxes pass an “at most 2” rule.
• A guesser answers 10 true/false questions (p = 0.5). The chance of getting at least 8 right is the upper tail, pbinom(7, 10, 0.5, lower.tail = FALSE) ≈ 5%, so scoring 8+ by luck alone is unlikely.

Frequently asked questions

Does pbinom include k?

Yes — pbinom(k) = P(X ≤ k). For P(X < k) use pbinom(k − 1).

How do I get P(at least k)?

Use pbinom(k − 1, lower.tail = FALSE), which is P(X ≥ k).