When flipping a coin, what is the probability of it landing on the same result 3 times in a row?

Understand the Problem

The question is asking for the probability of a coin landing on the same side (either heads or tails) three times in a row after being flipped. This involves basic probability concepts regarding independent events.

Answer

The probability is $P = \frac{1}{4}$.

Steps to Solve

Understand the probability of one flip
A fair coin has two possible outcomes for each flip: heads (H) or tails (T). The probability of landing on heads is $P(H) = \frac{1}{2}$ and the probability of landing on tails is $P(T) = \frac{1}{2}$.
Calculate the probability of three consecutive flips
Since each coin flip is independent, we multiply the probabilities for three flips. The probability of getting heads three times is:

$$ P(HHH) = P(H) \times P(H) \times P(H) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} $$

Similarly, the probability of getting tails three times is:

$$ P(TTT) = P(T) \times P(T) \times P(T) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} $$

Sum the probabilities
The total probability of landing on the same side (either all heads or all tails) three times is the sum of the probabilities of these two outcomes:

$$ P(\text{same side}) = P(HHH) + P(TTT) = \frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4} $$

The probability of a coin landing on the same result three times in a row is $P = \frac{1}{4}$.

More Information

This probability indicates that there is a 25% chance for a fair coin to land on the same side (either heads or tails) for three consecutive flips. This problem involves basic probability principles of independent events.

Tips

Forgetting that flips are independent: Each flip does not affect the others; thus, their probabilities multiply.
Not accounting for both outcomes: Only calculating one side (either heads or tails) can lead to an incomplete answer.

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