dbinom() — Exact binomial probability

The chance of getting exactly x wins out of n tries.

dbinom(x, size, prob)

Key idea

• dbinom gives the probability — the chance — of EXACTLY x successes in n independent tries (a binomial setup). (e.g. dbinom(2, 10, 0.5) = 0.044)
• Each try has the same success chance p, like a fair coin landing heads. (e.g. p = 0.5)
• Use it when you care about one specific count, not a range. (e.g. exactly 3 of 5)
• The most likely number of wins sits near n times p. (e.g. 10 x 0.5 = 5)
• The chances across all x from 0 to n add up to 1. (e.g. sum = 1)

Definition

$$P(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$$

$x$

number of successes

$n$

number of trials (size)

$p$

chance of success each trial

$\left(\genfrac{}{}{0ex}{}{n}{x}\right)$

ways to choose x of n

$$P(X=2)=\left(\genfrac{}{}{0ex}{}{10}{2}\right)(0.5{)}^2(0.5{)}^8=45\cdot \frac{1}{1024}=0.0439$$

Where you'd use it

• You flip a fair coin 10 times and wonder about getting exactly 3 heads. dbinom(3, 10, 0.5) ≈ 12%, so exactly 3 heads happens about one time in eight.
• A player sinks 80% of free throws. The chance they make exactly 7 of 10 is dbinom(7, 10, 0.8) ≈ 20% — sizeable, though 8 of 10 is actually their most likely count at about 30%.
• In a box of 20 phones where each has a 5% chance of being faulty, the chance of exactly 2 duds is dbinom(2, 20, 0.05) ≈ 19%.

Frequently asked questions

dbinom or pbinom?

dbinom for exactly k successes; pbinom for a cumulative "at most / at least".

What are size and prob?

size = number of trials, prob = chance of success on each trial.