Problem 61 - 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 on the \(k\) th trial for \(k=1,2\), and 3 .

Step-by-Step Solution

Answer

The probabilities are \( \frac{1}{2} \) for 1st trial, \( \frac{1}{4} \) for 2nd trial, and \( \frac{1}{8} \) for 3rd trial.

Step 1: Understand the Problem

This is a geometric probability problem, where we keep flipping a fair coin until heads appear for the first time. We are to calculate the probability that this first occurrence of heads happens specifically on the 1st, 2nd, and 3rd flip.

Step 2: Determine the Probability for Each Trial

The probability of getting heads on any individual flip of the fair coin is 0.5. It is a memoryless process, meaning past flips do not affect future outcomes.

Step 3: Probability for 1st Trial

For heads to appear on the 1st trial, it simply means getting heads on the first flip. The probability for this is directly:\[ P(\text{Heads on 1st trial}) = \frac{1}{2} \]

Step 4: Probability for 2nd Trial

For heads to appear on the 2nd trial, the first flip must be tails and the second flip must be heads. This sequence's probability is:\[ P(\text{Tails on 1st trial and Heads on 2nd trial}) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

Step 5: Probability for 3rd Trial

For heads to appear on the 3rd trial, the first two flips must be tails, and the third must be heads. The probability for this sequence is:\[ P(\text{Tails on 1st and 2nd trials and Heads on 3rd trial}) = \left(\frac{1}{2}\right)^3 = \frac{1}{8} \]

Step 6: Conclusion

We have calculated the probability for heads appearing first on the 1st, 2nd, and 3rd flip separately. They are \( \frac{1}{2} \), \( \frac{1}{4} \), and \( \frac{1}{8} \) respectively.

Key Concepts

Probability Theory

Probability theory is a fundamental branch of mathematics used to analyze random phenomena. When it comes to understanding probabilities, two important aspects are the likelihood of an event and how outcomes are measured.
Probability is often expressed as a number between 0 and 1—with 0 indicating impossibility and 1 indicating certainty. The probability of an event can often be expressed by the formula:

• \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

In the context of our coin-flipping scenario, the probability of getting heads on a single flip is 0.5 or \( \frac{1}{2} \), as the fair coin offers two equally likely outcomes: heads or tails.
For events occuring independently, like repeated coin tosses, the probability of a sequence of events is the product of individual probabilities.
This simple yet powerful concept allows us to calculate the probability of more complex sequences, such as getting heads for the first time on specific trials.

Random Experiments

A random experiment is any process that leads to one of several possible outcomes, where the results cannot be predicted with certainty in advance. In probability, these experiments are important because they model real-world events that have some inherent randomness.
Key components of a random experiment include:

• The sample space (\( S \)), which is the set of all possible outcomes.
• Events, which are subsets of the sample space.
• Probability, which assigns a likelihood to each event in the sample space.

In our context, flipping a coin repeatedly until heads appears is a random experiment. The sample space for each flip consists of {Heads, Tails}, and we're interested in sequences like {Heads}, {Tails, Heads}, {Tails, Tails, Heads}, etc.
This randomness is essential for calculating the probabilities of specific sequences, like getting the first head on the first, second, or third trial.

Memoryless Process

A memoryless process is an important concept in probability theory, where the outcomes of previous events do not affect the probabilities of future outcomes. This property makes sequences easier to analyze by ensuring that every trial is independent.
For a memoryless process, the chance of an event occurring remains constant, regardless of any previous occurrences. This is formally expressed in terms of conditional probability:

• \( P(A | B) = P(A) \)

if event \( B \) has already occurred, and does not change the probability of \( A \).
Coin flipping is a classic example of a memoryless process. Each flip remains independent of the others; thus, the probability of getting heads remains \( \frac{1}{2} \) on every flip. This key characteristic allows us to consider each trial separately when calculating probabilities, such as finding heads for the first time on the 1st, 2nd, or 3rd trial.