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 0.125.

Step 1: Understanding the Problem

This problem involves a random experiment of flipping a fair coin until a head appears. We want to calculate the probability that the first heads appears after the third flip.

Step 2: Defining the Event

We denote by "H" a heads result, and by "T" a tails result. The event where the first heads appears after the third trial corresponds to the outcome where the first three flips are tails, represented as TTT.

Step 3: Calculating Individual Probabilities

Since the coin is fair, the probability of getting a tail or a head on any flip is 0.5. Thus, the probability of a tail (T) on one flip is \( P(T) = 0.5 \).

Step 4: Probability of Tails on First Three Flips

Since each flip is independent, we calculate the probability that all first three flips are tails by multiplying their individual probabilities: \( P(T \text{ on 1st flip}) \times P(T \text{ on 2nd flip}) \times P(T \text{ on 3rd flip}) = 0.5 \times 0.5 \times 0.5 \).

Step 5: Solving the Multiplication

Perform the multiplication to find the probability: \( 0.5 \times 0.5 \times 0.5 = 0.125 \).

Step 6: Interpreting the Result

The probability that the first heads appears after the third trial (which is equivalent to the first three flips being tails) is 0.125.

Key Concepts

Understanding Random Experiments

In probability theory, a random experiment is a process or procedure that generates outcomes which are not predictable in advance. However, the set of possible outcomes is known.
For example, consider the simple act of tossing a coin. Before doing so, we cannot predict whether it will land heads or tails. Yet, we know these are the only possible outcomes. This makes coin flipping a classic example of a random experiment.

• Each outcome is equally likely.
• The experiment can be repeated multiple times.

Another key point is that the outcomes themselves are undefined until the experiment is executed, thus introducing an element of chance.
In our exercise, the random experiment is the process of flipping the coin until heads appears. The goal is to assess the likelihood of observing a particular sequence of outcomes (tails) before the first heads.

Exploring Independent Events

Independent events are a fundamental concept in probability. Two events are independent if the occurrence of one does not affect the occurrence of the other.
In a coin toss, the result of one flip does not influence the outcome of the next flip. This independence has a significant implication: the probabilities of independent events occurring in sequence can be calculated by multiplying their individual probabilities.

• Probability of heads: 0.5
• Probability of tails: 0.5

For example, the probability of getting tails on three consecutive flips of a fair coin is found by multiplying the probability of tails for each flip: 0.5 * 0.5 * 0.5 = 0.125.
This is because each flip is an independent event, unaffected by previous outcomes. Understanding this, we calculated the probability that the first heads appears after the third trial.

Significance of a Fair Coin

A fair coin is an unbiased coin, where each outcome (heads or tails) has an equal chance of occurring. This means the probability of landing heads is the same as landing tails, which is 0.5 for each.
This is crucial when analyzing probability in random experiments, as it ensures that each outcome is equally likely and not skewed in favor of one side.

• Equal probability of heads and tails ensures fairness.
• Calculations of outcomes become straightforward.

In our exercise, using a fair coin simplifies the probability calculations because each trial is the same. This regularity allows us to be confident that the chance of getting three tails before getting a head is exactly 0.125.
This underlines the concept of "fair" and ensures that each trial outcome is fair and square.