Question

What is the probability that a fair coin will come up heads three times in a row when tossed?

Short answer

The probability is \( \frac{1}{8} \).

Step-by-step solution

Understanding the Problem

We need to find the probability of getting heads three times in a row when tossing a fair coin. Each toss of a coin is an independent event with two equally likely outcomes: heads or tails.

Probability of One Toss

For each coin toss, the probability of getting heads is \( \frac{1}{2} \). This is because there are two possible outcomes: heads and tails, and they are equally likely.

Probability of Multiple Independent Events

To find the probability of multiple independent events all occurring, we multiply the probability of each individual event. Thus, the probability of getting heads on three successive tosses is \( \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \).

Calculate the Probability

Calculate \( \left( \frac{1}{2} \right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \). This is the probability of getting heads three times in a row with a fair coin.

Conclusion

Thus, the probability that a fair coin will come up heads three times in a row is \( \frac{1}{8} \).

Key concepts

Independent Events

In probability theory, an event is said to be independent if the outcome of one event does not affect the outcome of another. When we talk about independent events with coin tosses, it means that the result of one coin toss does not influence the next.

For example, if you toss a fair coin in the air, whether it lands on heads or tails has no bearing on what the next toss will land on. Each toss is a fresh start and is unaffected by previous outcomes.

Understanding this concept is crucial because it allows us to calculate the combined probability of multiple events. If you know the probability of one independent event, you can find the probability of it occurring multiple times by simply multiplying its probability by itself for each occurrence.

Coin Toss Experiment

A coin toss experiment is a classic way to illustrate probability concepts. It involves flipping a coin and observing the outcome—heads or tails.

This simple action lays the groundwork for understanding more complex probability issues. When flipping a fair coin, each side, heads or tails, is an equally likely outcome.

A single coin toss is a straightforward event, but when you consider sequences of tosses, you can explore patterns and probabilities of these patterns. For example, predicting the probability of a coin landing on heads multiple times in a row adds a layer of complexity, yet it remains grounded in simple principles of independent and repeated trials.

Equally Likely Outcomes

One of the core principles of probability is that each outcome in an experiment is equally likely, particularly in a fair situation like a coin toss. In the coin toss experiment, there are two possible outcomes: heads and tails.

When we say these outcomes are equally likely, it means there is a 50/50 chance for either. In mathematical terms, the probability of landing heads (or tails) on a single toss is \( \frac{1}{2} \) because there are 2 possible outcomes, both equally probable.

This assumption of equally likely outcomes underpins many probability calculations, leading to the predictable results when multiple independent events are considered, such as the scenario of tossing a coin multiple times. To solve this, you utilize this principle to assess how likely it is for a pattern, like three consecutive heads, to occur.