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 1/2 for k=1, 1/4 for k=2, and 1/8 for k=3.

Step 1: Define the Random Event

For a fair coin, there are two possible outcomes for each flip: heads (H) or tails (T). We are interested in finding the probability of getting our first heads on the \(k\)-th flip. This means the first \(k-1\) flips must all be tails, and the \(k\)-th flip must be heads.

Step 2: Calculate the Probability for the k-th Trial

Since each coin flip is independent, the probability of getting a tail is \(\frac{1}{2}\), and the probability of getting a head is \(\frac{1}{2}\). For the first heads to appear on the \( k \)-th flip, the sequence must be \( k-1 \) tails followed by a head. The probability of this specific sequence of \( k-1 \) tails and 1 head is given by the product: \[(\frac{1}{2})^{k-1} \times (\frac{1}{2}) = (\frac{1}{2})^k.\]

Step 3: Calculate Probabilities for Specific Values of k

Plug in \(k = 1\), \(k = 2\), and \(k = 3\) into the formula \((\frac{1}{2})^k\):- For \(k=1\), the probability \((\frac{1}{2})^1 = \frac{1}{2}.\)- For \(k=2\), the probability \((\frac{1}{2})^2 = \frac{1}{4}.\)- For \(k=3\), the probability \((\frac{1}{2})^3 = \frac{1}{8}.\)

Key Concepts

Random Experiment

A random experiment is a process that leads to one of several possible outcomes. In probability theory, it is often an uncertain or unpredictable process. For instance, when flipping a coin, there are two potential outcomes: heads or tails. This simple flip is a classic example of a random experiment.

Helpful insights:

• Random experiments are fundamental in understanding probability. They help model situations where outcomes can't be predetermined.
• The essence of a random experiment is its unpredictability; each trial's result doesn't influence the next.
• They provide a foundational block for constructing complex probability problems.

In our exercise, each flip of the coin is a random experiment. We're wondering when, during a series of flips, we might first get a heads. This type of problem is a typical scenario used to teach probability concepts.

Independent Events

Independent events occur when the outcome or occurrence of one event does not affect the outcome of another. When dealing with coin flips, each flip is considered an independent event.

Key points about independent events:

• Independence indicates that the result of one trial does not change the probability of the next trial’s results.
• For coins, every flip is separate. No matter how many times you flip the coin, previous results don't affect future ones.
• This property allows us to calculate probabilities using simple multiplication when determining sequences of results.

In our exercise, the phrase "each coin flip is independent" is crucial because it implies that the probability of getting tails, then tails, then heads (for instance) is simply the product of each individual flip's probability. This independence simplifies probability calculations.

Sequence Probability

Sequence probability is the probability of a series of events occurring in a specific order. In probability theory, you frequently calculate this by multiplying the probabilities of independent events.

Understanding sequence probability:

• Sequence probability focuses on the particular order of events needed for a desired outcome.
• When events are independent, the probability of a sequence of events is found by multiplying their individual probabilities together.
• This approach is used in our exercise, where we calculated the probability of getting exactly one head in a string of flips.

In the context of our problem, determining the chance that the first head appears on the \( k \)-th trial means calculating the probability of getting tails up to the \((k-1)\)-th trial and heads on the \( k \)-th. It's a great way to see how sequence probabilities unfold when events are independent.