Problem 65 - Calculus for Biology and Medicine

Question

A random experiment consists of flipping a fair coin until the first time heads appears. Find the probability that the first heads appears after the third trial.

Step-by-Step Solution

Answer

The probability is \( \frac{1}{16} \).

Step 1: Understanding the Event

To solve this problem, we need to calculate the probability of getting the first heads on the fourth trial. This implies that the first three trials result in tails and the fourth trial results in heads.

Step 2: Identifying Success and Failure Probabilities

Since the coin is fair, the probability of getting a tails on any single trial is \( P(T) = \frac{1}{2} \) and the probability of getting a heads is \( P(H) = \frac{1}{2} \).

Step 3: Defining the Event Probability

For the first three trials to result in tails, the probability is \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \).

Step 4: Calculating the Fourth Trial Probability

On the fourth trial, we want to get heads, so the probability for heads is \( \frac{1}{2} \).

Step 5: Combining Probabilities

The probability of getting three tails followed by a head is the product of these probabilities: \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \).

Step 6: Final Calculation

Thus, the probability that the first heads appears after the third trial is \( \frac{1}{16} \).

Key Concepts

Random Experiment

In probability theory, a random experiment is a fundamental concept that refers to any process for which the outcome cannot be predicted with certainty. It involves performing an activity that can produce different outcomes, each with an assignable probability. Random experiments are essential for understanding probability as they provide a structured way to analyze how likely different outcomes are.

When you flip a coin, an everyday example of a random experiment unfolds. You are unsure whether the result will be heads or tails until the process concludes. Each flip is an independent trial, contributing to the overall understanding of the experiment's probabilistic nature. In our specific exercise, the random experiment is flipping a fair coin until the first head appears. We repeat this flip process until a specific outcome, heads, is achieved, making it a classic example of a random experiment.

Key characteristics of a random experiment include:

• Uncertain outcomes.
• Multiple possible results.
• Repetition of trials under identical conditions.

Fair Coin

A fair coin is a critical component in probability exercises and experiments, signifying a coin for which both sides, heads and tails, have an equal chance of landing face up. This concept is crucial because it provides a foundation for using probability theory to make predictions about the outcomes of random experiments involving coin flips.

When we say a coin is fair, we mean that there is no bias toward one side or the other. The likelihood of landing on heads is equal to the likelihood of tails, both having a probability of \( \frac{1}{2} \) per flip. This implies that in a large number of flips, we would expect heads to appear about half of the time and tails the other half. Fair coins ensure that experiments involving coin flips remain unbiased, allowing for an accurate application of probability calculations.

In our exercise, using a fair coin simplifies probability calculations, ensuring the assumptions in our probability theory hold true throughout.

Probability Calculation

Probability calculation involves determining the likelihood of a particular outcome occurring within a random experiment. It is a fundamental aspect of probability theory that combines the concepts of random experiments and fair coins.

In the context of our exercise, the probability calculation helps us find how likely it is to get the first heads on the fourth coin flip. To achieve this, recognize the sequences of events required: three tails (indicating failures) followed by one head (indicating success). Each coin flip being independent means the probability of a tail is \( \frac{1}{2} \), and similarly, so is the probability of a head.

The probability of the first three flips resulting in tails is calculated as:

• \( \left(\frac{1}{2}\right)^3 = \frac{1}{8} \)

Following this, the probability of a head on the fourth flip is:

• \( \frac{1}{2} \)

These probabilities are multiplied together for the final probability calculation, giving:

• \( \frac{1}{8} \times \frac{1}{2} = \frac{1}{16} \)

Thus, the probability of the first heads appearing after the third trial is \( \frac{1}{16} \). Understanding this calculation helps illustrate the combined use of probability theory principles in evaluating random experiments.