The probability is 1/2 that a certain coin will turn...

Solving directly for the probability that we will get tails at least once is somewhat complicated — we have to think about the probability of getting tails on one, two, three, or four tosses, as well as in which order the tails will appear. However, we can solve this problem easily by instead finding the probability that we will get heads every single toss, then subtracting from one. This gives us the probability that we will NOT get heads every single toss — in other words, we will get tails at least once.

The probability we will get heads on each toss is 1/2. Multiplying by 1/2 for each toss gives
$$\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{16}$$
So the probability that we will get heads every single toss is 1/16.

Subtracting this from one gives
$$1-\frac{1}{16}=\frac{15}{16}$$
So the probability that we will get tails at least once is 15/16, or answer choice E.

When solving for the probability that something will occur, it's often easiest to subtract out the cases that won't occur. Not always the case, but useful to keep in your tool belt!