Changing binwidth of density histogram so that probabilities sum up to 1 - r

Question

I have already found numerous of questions to it, but somehow it did not really help me. I do not understand how to change the binwidth in a density histogram in ggplot2, so that the probabilities sum up to 1. It seems like it only works if the binwidth is exactly 1.
Here is an example:
set.seed(1)
df = data.frame("data" = runif(1000, min=0, max=100))
a = ggplot(data = df, aes(x = data))+
geom_histogram(aes(y=..density..),colour="black", fill = "white",
breaks=seq(0, 100, by = 50))
b = ggplot(data = df, aes(x = data))+
geom_histogram(aes(y =..density..),
breaks=seq(0, 100, by = 30),
col="black",
fill="white")
c = ggplot(data = df, aes(x = data))+
geom_histogram(aes(y =..density..),
breaks=seq(0, 100, by = 10),
col="black",
fill="white")
d = ggplot(data = df, aes(x = data))+
geom_histogram(aes(y =..density..),
breaks=seq(0, 100, by = 1),
col="black",
fill="white")
grid.arrange(a,b,c,d, ncol= 2)
If you look at the probability axis, you can see that the first three graphs must be wrong. These are not the right histograms as the bins do not sum up to 1. The y-axis even does not change significantly according to the histogram a, b, c or d. I also tried to replace the "breaks" command by the "binwidth" command, but it is even worse then.
I would also like to know how you can count the probabilities of the single bins of a histogram to proof that it sums up to 1 or not?
Thanks for any help.

Answer 1

Simulate some data:
library(ggplot2)
library(dplyr)
set.seed(1)
df = data.frame("data" = runif(1000, min=0, max=100))
The first plot you can get is:
# y axis has the density estimate values
ggplot(data = df, aes(x = data))+
geom_histogram(aes(y=..density..),colour="black", fill = "white",
breaks=seq(0, 100, by = 50))
This plot has the density estimates on the y axis. Those values correspond to the density plot and not to the bars you created. You can see this version where the density plot is overlayed:
# y axis has the density estimate values and the density plot
ggplot(data = df, aes(x = data))+
geom_histogram(aes(y=..density..),colour="black", fill = "white",
breaks=seq(0, 100, by = 50)) +
geom_density(aes(data), col="red")
A way to interpret this is that each point on the red line has a probability to be selected and that's on the y axis (i.e. lots of points means that probabilities tend closer to zero).
You can get what you want with this:
# y axis has the probabilities of each bar (bar counts / all counts)
ggplot(data = df, aes(x = data))+
geom_histogram(aes(y=..count../sum(..count..)),colour="black", fill = "white",
breaks=seq(0, 100, by = 50))
Another way to do the above, while keeping the data (for future usage or just check probabilities sum to 1) is this:
# assign the breaks
breaks = cut(df$data, seq(0, 100, by = 50))
# count observations in each bar and probability of each bar
df %>%
mutate(Breaks = breaks) %>%
count(Breaks) %>%
mutate(Prc = n/sum(n))
# # A tibble: 2 x 3
# Breaks n Prc
# <fctr> <int> <dbl>
# 1 (0,50] 520 0.52
# 2 (50,100] 480 0.48
# plot the above
df %>%
mutate(Breaks = breaks) %>%
count(Breaks) %>%
mutate(Prc = n/sum(n)) %>%
ggplot(aes(Breaks, Prc)) + geom_col()