An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is:

We are asked to find the probability that, when an unbiased coin is tossed eight times, we get at least one Head and at least one Tail.

First, recall the basic probability formula: if all outcomes are equally likely, then

Probability $$=\frac{$$ Number of favourable outcomes $$}{$$ Total number of possible outcomes $$}.$$

However, counting directly the “favourable outcomes” (all sequences that contain both Heads and Tails) is lengthy. A common technique is to use the Complement Rule, which states:

$$P($$ Event $$) = 1 - P($$ Complement of the event $$).$$

Here the event we want is “at least one Head and at least one Tail.” Its complement is “no Head or no Tail,” which translates to “all Heads” or “all Tails.”

Now, let us determine the total number of possible outcomes when a coin is tossed eight times. Each toss has 2 possible results (H or T). Hence, by the rule of product, the total number of sequences of length 8 is

$$2^8 = 256.$$

Because the coin is unbiased, every sequence is equally likely, so the probability of any single sequence is

$$\frac{1}{256}.$$

Next, we count the outcomes in the complement:

• All eight tosses are Heads: there is exactly one such sequence, namely HHHHHHHH.
• All eight tosses are Tails: again, there is exactly one sequence, namely TTTTTTTT.

Thus, the number of complementary outcomes is

$$1 + 1 = 2.$$

Therefore, the probability of the complement event (“all Heads or all Tails”) is

$$P(\text{all Heads or all Tails}) = \frac{2}{256}.$$

We can simplify this fraction step by step:

First divide numerator and denominator by 2:

$$\frac{2}{256} = \frac{1}{128}.$$

Now we apply the Complement Rule:

$$\begin{aligned} P(\text{at least one Head and at least one Tail}) &= 1 - P(\text{all Heads or all Tails}) \\[4pt] &= 1 - \frac{1}{128}. \end{aligned}$$

We convert the whole number 1 to a fraction with denominator 128 to combine easily:

$$1 = \frac{128}{128}.$$

Subtracting term-by-term, we obtain

$$\frac{128}{128} - \frac{1}{128} = \frac{127}{128}.$$

So the required probability is

$$\frac{127}{128}.$$

Hence, the correct answer is Option A.