The probability distribution for 350 successes out of 1,000 trials is much more sharply peaked than the distribution of 35 successes out of 100 trials, even though they both represent a 35% success rate.

The formula for the cumulative distribution function is:

You can calculate this directly in R using the code below. The `choose()`

function is the \binom{n}{k} in the formula above. It represents the number of ways to choose *k* out of *n* elements.

```
sum(choose(100, 0:35) * 0.5^(0:35) * 0.5^(100:65))
#> [1] 0.001758821
sum(choose(1000, 0:350) * 0.5^(0:350) * 0.5^(1000:650))
#> [1] 8.078155e-22
```

^{Created on 2018-12-06 by the reprex package (v0.2.1)}

In the examples below, note how the number of ways to choose *k* out of *n* elements becomes more peaked as *n* increases. The vertical axis is \binom{n}{k} for each *k*:

```
par(mfrow=c(3,1))
plot(choose(10, 0:10), pch=16, cex=0.7, las=1, ylab="", xlab="k")
plot(choose(100, 0:100), pch=16, cex=0.7, las=1, ylab="", xlab="k")
plot(choose(1000,0:1000), pch=16, cex=0.7, las=1, ylab="", xlab="k")
```

And here's a graphical example of the cumulative distribution function from your example:

```
plot(0:100/100, pbinom(0:100, 100, 0.5), type="o", pch=16)
lines(0:1000/1000, pbinom(0:1000, 1000, 0.5), col="red")
points(0:1000/1000, pbinom(0:1000, 1000, 0.5), col="red", pch=16)
```