A standard fair coin is tossed three times. What is the probability that the three outcomes are all the same?
(A) $\frac{1}{2}$
(B) $\frac{3}{16}$
(C) $\frac{1}{4}$
(D) $\frac{5}{16}$
(E) $\frac{1}{8}$
Solution 1

When a standard fair coin is tossed three times, there are 8 possible outcomes.

If $\mathrm{H}$ represents head and $\mathrm{T}$ represents tail, these 8 outcomes are: HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.

Of these 8 outcomes, there are exactly 2 whose outcomes are all the same (HHH and TTT).

Therefore, the probability that the three outcomes are all the same is $\frac{2}{8}=\frac{1}{4}$.

## Solution 2

The three outcomes are all the same exactly when each of the three tosses is a head or when each of the three tosses is a tail.

When a coin is tossed, the probability that the coin shows a head is $\frac{1}{2}$ and the probability that the coin shows a tail is also $\frac{1}{2}$ (there are two possible outcomes and each is equally probable). Thus, the probability that each of the three tosses is a head is $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{8}$.

Similarly, the probability that each of the three tosses is a tail is $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}=\frac{1}{8}$.

Therefore, the probability that the three outcomes are all the same is $\frac{1}{8}+\frac{1}{8}=\frac{2}{8}=\frac{1}{4}$.

ANSWER: (C)