Binomial cumulative probabilities and pbinom

Add the pmf up to k; "at least" by the complement; and what dbinom and pbinom return — the "more than" trap.

Building the cumulative table

For $X \sim Bin (12, 0.15)$ the pmf values, from L6.4's formula:

$x$	0	1	2	3	4
$p (x)$	0.1422	0.3012	0.2924	0.1720	0.0683
$P (X \leq x)$	0.1422	0.4435	0.7358	0.9078	0.9761

The second row is the running total — the cumulative probability. Any phrase from L6.1 is now one or two lookups:

At most 2 errors: $P (X \leq 2) = 0.1422 + 0.3012 + 0.2924 = 0.7358$ .
At least 3: $P (X \geq 3) = 1 - P (X \leq 2) = 1 - 0.7358 = 0.2642$ . Ten terms by the direct route; one subtraction by the complement.
More than 3: $P (X > 3) = 1 - P (X \leq 3) = 1 - 0.9078 = 0.0922$ .
Between 1 and 3 inclusive: $P (X \leq 3) - P (X \leq 0) = 0.9078 - 0.1422 = 0.7656$ .

Every cumulative question reduces to which $k$ goes into $P (X \leq k)$ and whether to subtract from 1. That is the whole skill.

What R prints

The paper may hand you R output instead of a table. Two functions matter:

> dbinom(2, 12, 0.15)
[1] 0.2923585
> pbinom(2, 12, 0.15)
[1] 0.7358181
> 1 - pbinom(2, 12, 0.15)
[1] 0.2641819

dbinom(x, n, p) is the pmf, $P (X = x)$ — the d is for density, the name R uses for both kinds of variable. pbinom(k, n, p) is the cumulative $P (X \leq k)$ — less than or equal, always. So 1 - pbinom(2, …) is $P (X \geq 3)$ , and the three printed lines are the three hand values above at seven significant figures.

The trap

"More than 2" is not 1 - pbinom(2, …). That expression is $P (X > 2) = P (X \geq 3)$ — which is more than 2. The trap runs the other way: "at least 2" is $1 - P (X \leq 1)$ = 1 - pbinom(1, …), not 1 - pbinom(2, …). Translate the phrase to an inequality, move the boundary so the right side is $P (X \leq k)$ , then write the call:

Phrase	Inequality	R
at most $k$	$P (X \leq k)$	`pbinom(k, n, p)`
fewer than $k$	$P (X \leq k - 1)$	`pbinom(k - 1, n, p)`
more than $k$	$1 - P (X \leq k)$	`1 - pbinom(k, n, p)`
at least $k$	$1 - P (X \leq k - 1)$	`1 - pbinom(k - 1, n, p)`

R also accepts lower.tail = FALSE, which returns $P (X > k)$ directly — the same as 1 - pbinom(k, …); you will meet it in Module 7 with pnorm.

Reading the seven digits

R prints 0.7358181 where the hand table says 0.7358. Both are right; the paper's tolerance (±0.0005 for a four-decimal pmf sum) accepts either. Never quote more decimals than the question's own inputs support.

The cumulative table for Bin(12, 0.15). P(X ≤ 2) = 0.7358 is what pbinom(2, 12, 0.15) returns; P(X ≥ 3) = 1 − 0.7358.