What do you notice about the probability of getting zero hea

What do you notice about the probability of getting zero heads and of getting four heads. Similarly for one head and three heads. Can you make an argument for why the states of zero heads and all heads have the same probability of occurring and the states one head and all-but one head have the same probabilities without any reference to counting microstates?

Solution

The probability of heads is the same as the probability of tails in a coin toss.

Also, note that C(n, r) = C(n, n-r).

Thus, by these symmetries, the described probabilities in the problem are equal.