Problem 67 - 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 within the first four trials.

Step-by-Step Solution

Answer

The probability is \( \frac{15}{16} \).

Step 1: Define the Probability of Each Flip

Since the coin is fair, the probability of getting a heads (H) on each flip is \( \frac{1}{2} \), and the probability of getting tails (T) is also \( \frac{1}{2} \).

Step 2: Identify the Desired Outcomes

The event "first heads within four trials" means getting at least one heads in the first four flips. The possibilities are: H, TH, TTH, TTTH.

Step 3: Calculate the Probability for Each Outcome

- **Outcome 1 (H)**: H appears on the first trial. Probability is \( \frac{1}{2} \).- **Outcome 2 (TH)**: First flip T, then H. Probability is \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \).- **Outcome 3 (TTH)**: First two flips T, then H. Probability is \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \).- **Outcome 4 (TTTH)**: First three flips T, then H. Probability is \( \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16} \).

Step 4: Sum the Probabilities of all Desired Outcomes

Sum the calculated probabilities from Outcomes 1, 2, 3, and 4:\[ P(\text{First heads within four trials}) = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} \]

Step 5: Simplify the Expression

Convert all fractions to have the same denominator and sum:\[ \frac{8}{16} + \frac{4}{16} + \frac{2}{16} + \frac{1}{16} = \frac{15}{16} \]

Step 6: Conclusion: Probability Found

The probability that the first heads appears within the first four trials is \( \frac{15}{16} \).

Key Concepts

Understanding Random Experiments

When we dive into the world of probability, a key term we encounter is "random experiment." A random experiment is any activity or process that leads to one or more possible outcomes, where the result cannot be precisely predicted beforehand.
In simpler terms, it's like a test where we know the potential outcomes, but not which one will actually occur on any given trial.
For instance, flipping a coin is considered a random experiment because there are two possible outcomes—getting a head (H) or a tail (T)—and we can't control or predict the result of a specific flip.

The Concept of a Fair Coin

In our exercise, we are flipping a "fair coin." But what does "fair" actually mean? In probability terms, a fair coin is one that has equal chances for landing on heads or tails.
Every flip is independent, meaning the result of one flip doesn't impact the next.
Both outcomes, head or tail, have a probability of 0.5, or equivalently, \( \frac{1}{2} \) each.

• This is because a fair coin should have no bias towards either side.
• Thus, every flip remains a 50/50 chance.

With this balance, each trial in our experiment remains consistent in terms of chance, reflecting the unpredictability of real-world events.

How to Calculate Probability

Understanding probability calculation is crucial when predicting outcomes of random experiments. To calculate the probability of an event, you compare the number of successful outcomes to the total possible outcomes.
In our exercise, we aim to determine the probability of seeing the first head within four coin flips.
Let’s break it down:

• The possible outcomes to consider are: heads on the first, second, third, or fourth flip.
• Each scenario has its specific probability; for instance, getting heads on the first flip is \( \frac{1}{2} \), while on the second flip (after a tail) is \( \frac{1}{4} \) because it combines two independent flips.

By summing these probabilities, we find the overall likelihood of our desired event occurring within the first four flips.

Trial Outcomes in Probability

Trial outcomes are the potential results we examine in a random experiment.
For the fair coin, each flip constitutes a trial.
In our coin flip exercise:

• An outcome might be the sequence H (head on first flip), TH (head on second), etc.
• Identifying each outcome's probability is essential for computing the overall event probability.

Each outcome combines independent events (flips). We calculate each scenario's probability by multiplying the probabilities of individual flips. Summing these probabilities gives us the total probability that the first head appears within four trials.