Question

If a fair coin is flipped three times, what is the probability of getting heads, tails, heads, in that order? A. \(\frac{1}{16}\) B. \(\frac{1}{8}\) C. \(\frac{1}{4}\) D. \(\frac{1}{2}\)

Short answer

B. \(\frac{1}{8}\)

Step-by-step solution

- Determine the Probability of Each Flip

A fair coin has two possible outcomes for each flip: heads (H) or tails (T). The probability of getting heads in one flip is \(\frac{1}{2}\) and the probability of getting tails in one flip is also \(\frac{1}{2}\).

- Calculate the Combined Probability

For independent events, the probabilities are multiplied. We need H on the first flip, T on the second flip, and H on the third flip. The combined probability is: \ \[ P(H, T, H) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

- Choose the Correct Answer

The calculated probability \(\frac{1}{8}\) matches one of the options provided. Therefore, the correct answer is B. \(\frac{1}{8}\).

Key concepts

Independent Events

In probability, independent events are those whose outcomes do not affect each other. This means that the outcome of one event has no impact on the outcome of another. For example, when flipping a fair coin multiple times, each flip is an independent event because the result of one flip (heads or tails) does not influence the result of the next flip. Therefore, the probability of each flip remains consistent throughout the sequence.

Coin Flips Probability

When dealing with coin flip probability, each coin flip has only two possible outcomes: heads (H) or tails (T). Because we are dealing with a fair coin, the probability of getting heads is always \(\frac{1}{2}\) and the probability of getting tails is also \(\frac{1}{2}\). This stays the same no matter how many times we flip the coin. In the given exercise, when we want to find the probability of getting a specific sequence like H, T, H, we need to understand that each flip is an independent event, so each has a \(\frac{1}{2}\) chance of occurring.

Combined Probability

To find the probability of having a sequence of events occur, such as flipping a coin three times in order to get heads, tails, and heads, we multiply the probabilities of each individual event. This is because, for independent events, the combined probability is the product of the individual probabilities. For example, the probability of flipping heads (H), tails (T), and heads (H) in that specific sequence is calculated by multiplying each individual probability together: \[ P(H, T, H) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]. This is why the probability of getting H, T, H in that order is \(\frac{1}{8}\). This principle can be applied to any sequence of independent events.