What is the probability that both heads and tails occur?

Question

Assume that the coin is weighted so that a tail is 6 times as likely as a head. The coin is flipped 7 times.

Answer 1

I am assuming that the question means "What is the probability that we have at least one head AND at least one tail".

The easiest way to do this is to realize that we are only excluding two situations: ALL heads and ALL tails. The probabilities of those are (P(head))^7 and (P(tail))^7 and therefore our answer needs to be 1 - (P(head))^7 - (P(tail))^7.

Tail is six times as likely as head. So: 6P(head) = P(tail), and of course P(tail) = 1 - P(head), so 6P(head) = 1 - P(head) and we end up with P(head) = 1/7, P(tail) = 6/7.

And so, we have 1 - (P(head))^7 - (P(tail))^7 = 1 - (1/7)^7 - (6/7)^7 = .66008 . This is the probability of a result that does not have all heads and does not have all tails.