Question

If you flipped a fair coin three times, what is the probability that it would come up "heads" at least once? %

Answer 1

If you flip the coin three times, you have four possible number of heads:

A: No heads. all coins land on tails

B: One heads. one coin lands on heads, two coins land on tails

C: Two heads. two coins land on heads, one coin lands on tails

D: Four heads. all coins land on heads.

You're interested in the union of the latter three events (one, two or all heads). In this case, it is smarter to compute the probability of the remaining event (no heads), and compute the probability of the other three by completing to 1: we have

$P(A) + P(B) + P(C) + P(D) = P(A\cup B\cup C\cup D) = 1 \implies P(B\cup C \cup D) = P(B) + P(C) + P(D) = 1-P(A)$

We have $P(A) = 0.5^3$, because we want three consecutive flips to land on tails, and each flip has a probability of 0.5 to land on tails.

So, at least one coin lands on heads with probability $1-0.5^3$