Question

A fair coin is one that has probability \(1 / 2\) of coming up heads when flipped. a. What is the probability that a fair coin will come up tails \(n\) times in a row? b. Find the probability that a coin comes up heads for the first time after an even number of coin flips.

Short answer

a. Probability is \( \left( \frac{1}{2} \right)^n \). b. Probability for even number is \( \frac{1}{3} \).

Step-by-step solution

Understanding the Probability of Tails

When flipping a fair coin, the probability of getting tails on a single flip is \( \frac{1}{2} \). To determine the probability of getting tails \( n \) times in a row, we need to multiply the probability of getting tails on each flip together. Therefore, the probability of getting tails \( n \) times in a row is \( \left( \frac{1}{2} \right)^n \). This arises because the flips are independent events. A quick example: if \( n = 3 \), the probability is \( \left( \frac{1}{2} \right)^3 = \frac{1}{8} \).

Calculating Probability for Even Flip Number

To find the probability that a coin comes up heads for the first time after an even number of coin flips, note that for the first \( n \) flips (where \( n \) is even), you must get tails, and then heads on the \( n+1 \)-th flip. For an even \( n \, = 2k \), the probability of this scenario is \( \left( \frac{1}{2} \right)^{2k} \cdot \frac{1}{2} = \left( \frac{1}{2} \right)^{2k + 1} \). Since \( k \) can be any positive integer, we sum the probabilities for all even \( n \): \( \sum_{k=1}^{\infty} \left( \frac{1}{2} \right)^{2k + 1} \). This series is geometric with a common ratio \( r = \frac{1}{4} \) starting from \( \left( \frac{1}{2} \right)^{3} \).

Solving the Geometric Series

The sum of an infinite geometric series with first term \( a \) and common ratio \( r \) (where \(|r| < 1\)) is given by \( \frac{a}{1 - r} \). Here, the first term \( a = \left( \frac{1}{2} \right)^{3} = \frac{1}{8} \) and the common ratio \( r = \frac{1}{4} \). So, the sum is \( \frac{\frac{1}{8}}{1 - \frac{1}{4}} = \frac{\frac{1}{8}}{\frac{3}{4}} = \frac{1}{6} \). This means that the probability of the first head occurring after an even number of flips is \( \frac{1}{3} \).

Key concepts

Independent Events

In probability, we often encounter situations involving more than one random event. When these events do not affect each other, we refer to them as independent events. A classic example is the flipping of a fair coin. Each flip has no influence on others: the outcome of one flip (heads or tails) doesn't change the probability of the same result happening next time.

This independence becomes crucial when calculating probabilities over multiple trials. For instance, if you want to find the probability of getting tails three times in a row, you multiply the probability of tails for each independent flip. For a fair coin, with tails' probability as \( \frac{1}{2} \), the calculation goes: \( \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) \times \left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \). This multiplication highlights the independence of each event.

Remember, independence is a key concept that simplifies calculating probabilities in many scenarios. It allows us to analyze complex events by breaking them down into simpler, manageable parts.

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." Understanding geometric series helps in solving problems involving repeated experiments like coin flipping until a desired outcome occurs.

For our coin example, say we wish to find the probability that the first head appears after an even number of flips. This situation can be modeled as a geometric series because each sequence of tails followed by heads represents a repeated trial pattern. Specifically, if heads appear after 2, 4, 6, ... trials, each has a probability calculated as powers of \( \frac{1}{2} \).

• First term \( a = \left( \frac{1}{2} \right)^3 = \frac{1}{8} \)
• Common ratio \( r = \left( \frac{1}{4} \)

The infinite geometric series formula \( \frac{a}{1 - r} \) helps sum these terms to find total probability. Here, \( \frac{\frac{1}{8}}{1 - \frac{1}{4}} = \frac{1}{6} \). So, heads first appearing after an even count of flips has a probability of \( \frac{1}{3} \). Understanding this pattern is essential for predicting events over repeated trials.

Fair Coin

A fair coin is an idealized coin where each of its two outcomes, heads or tails, happens with equal likelihood on every flip. This means each side has a probability of \( \frac{1}{2} \). This balanced nature of a fair coin makes it a perfect model for studying basic probability concepts.

Using a fair coin helps demonstrate the principle of expected outcomes over many trials. When flipped repeatedly, you expect not only each outcome to appear approximately half the time due to its uniform probability but also to explore scenarios involving consecutive results, such as a sequence of tails.

The concept of a fair coin ties into understanding randomness and fairness in probability. While each trial is independent, the cumulative results should reveal these balanced odds, illustrating core principles of probability theory.