Question

need help understanding how expected frequencies are calculated from 4 coins tossed

Answer 1

given number of heads number of observed
0 8
1 40
2 50
3 40
4 22
find expected frequencies of each possible number of heads, in repetitions of this experiment

Answer 2

I suspect you mean, how does one calculate the theoretical frequencies of getting 0, 1, 2, 3 and 4 heads from four tosses of a fair coin?

1) The simpler way of seeing this is to draw, or imagine, a tree like the one attached where each branch corresponds to an outcome of one toss (except that I have shown all of the branches in full). There are 16 outcomes with the same probability, 1/16. To find the probability of obtaining three heads one would count the branches with exactly three heads; then the probability of the outcome would be this number multiplied by 1/16. The experiment has been repeated 160 (if I have added correctly). Then the expected number of trials with no heads would be 160/16 = 10, which compares closely to the 8 seen experimentally.

2) These probabilities are given by the binomial distribution (see wikipedia) probability mass function (pmf). For the pmf and theoretical frequencies I obtain.

(0.0625, 10.0)
(0.25000000000000006, 40.000000000000007)
(0.37500000000000011, 60.000000000000014)
(0.25000000000000006, 40.000000000000007)
(0.0625, 10.0)