Shape of a Histogram with Equally Likely Outcomes In a large class of introductory Statistics students, the professor has each student toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a Histogram of these proportions. What shape would you expect the Histogram to be? Why?
The answer says: We would expect the Histogram to be uni-modal and symmetric (i.e., shaped like a Normal curve) because the probability of heads and tails are equal.
Why does the fact that the probabilities of heads and tails are equal imply a uni-modal and symmetric Histogram?

If there are 100 students in the class and they are tossing fair coins independently, then you have
a sample of size 100 from $\mathsf{Binom}(16, 0.5).$
Simulating this 100-student experiment in R, I get
the histogram shown.
set.seed(129)
x = rbinom(100, 16, .5)
cutp = (0:17)-.5
hdr = "100 observations from BINOM(16,.5)"
hist(x, prob=T, br=cutp, col="skyblue2", main=hdr)


The following copy of this histogram has red dots showning expected
probabilities for each value in the histogram.
(One hundred observations are not enough for the histogram bars to be close to their expected heights.)
k = 0:16;  pdf = dbinom(k, 16, .5)
hist(x, prob=T, br=cutp, col="skyblue2", main=hdr)
points(k, pdf, pch=19, col="red")


If this were a really large class of $10\,000$
online students, then (after tabulation of all the results) the figure might look like
this:
set.seed(2022)
y = rbinom(10^4, 16, .5)
cutp = (0:17)-.5
hdr = "10,000 observations from BINOM(16,.5)"
hist(y, prob=T, br=cutp, col="skyblue2", main=hdr)
 points(k, pdf, pch=19, col="red")


