Problem 68 - Calculus for Biology and Medicine

Question

A random experiment consists of rolling a fair die until the first time a 1 or a 2 appears. Find the probability that the first 1 or 2 appears within the first five trials.

Step-by-Step Solution

Answer

The probability is \( \frac{211}{243} \).

Step 1: Understanding the Probability of Rolling a 1 or 2

Each roll of a fair six-sided die has an equal probability of landing on any of the six faces. Therefore, the probability of rolling a 1 or 2 in a single roll is the sum of the probabilities of rolling a 1 or rolling a 2, that is, \( \frac{1}{6} \) for a 1 and \( \frac{1}{6} \) for a 2. So the probability is \( \frac{1}{6} + \frac{1}{6} = \frac{1}{3} \).

Step 2: Calculating the Complement Probability

To find the probability that the first 1 or 2 appears within the first five trials, we should first calculate the probability of not rolling a 1 or 2 in a single trial. The probability of not rolling a 1 or 2 is then \( 1 - \frac{1}{3} = \frac{2}{3} \).

Step 3: Probability of No 1 or 2 in Five Rolls

Next, we determine the probability of not rolling a 1 or 2 over five consecutive rolls. This is \( \left( \frac{2}{3} \right)^5 = \frac{32}{243} \).

Step 4: Finding the Probability of at Least One 1 or 2 in Five Rolls

Using the complement rule, the probability of getting at least one 1 or 2 in five rolls is the complement of not getting any 1 or 2 in those five rolls. Thus, the probability is \( 1 - \frac{32}{243} = \frac{211}{243} \).

Step 5: Conclusion

The probability that the first occurrence of a 1 or 2 appears within the first five trials is \( \frac{211}{243} \).

Key Concepts

Random Experiment

A random experiment is a process that leads to an outcome that cannot be predicted with absolute certainty. For example, tossing a coin or rolling a die are both random experiments. Each time you perform the experiment, the outcome may differ.
When you roll a six-sided die, you are conducting a random experiment because the result could be any of the numbers from 1 to 6 with equal likelihood. In the context of our exercise, the random experiment is rolling the die until either a 1 or a 2 appears.
This means repeating the die roll until achieving a specific outcome, adding a layer of chance to each attempt. Therefore, each roll is independent of the previous rolls, meaning the outcome of one roll does not affect the next one. It is the foundation of many probability calculations, as these experiments provide different possible outcomes which we analyze to determine likelihoods.

Complement Rule

The complement rule is a fundamental concept in probability. It involves the use of complements to simplify the calculation of probabilities.
Essentially, it states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. This is powerful in situations where calculating the direct probability is challenging but finding its complement is easier.
In our die rolling scenario, instead of directly calculating the probability of rolling a 1 or 2 within five trials, we first determine the probability of NOT rolling either a 1 or 2 in five trials.
The complement rule then allows us to subtract this probability from 1 to find the probability of the event of interest, which is rolling at least one 1 or 2 in the five trials. This approach is often simpler and more efficient, especially in complex experiment setups.

Fair Six-Sided Die

Understanding the term 'fair six-sided die' is crucial for calculating probabilities in dice-related random experiments. A fair six-sided die means that each of the six faces has an equal probability of facing up when the die is rolled.
This ensures that each side, numbered 1 through 6, has a \(\frac{1}{6}\) probability of landing face up in a single roll.
The concept of a fair die is important because it forms the basis of the probability calculations in our exercise. Without fairness, the probability of certain numbers appearing would not be equal, making calculations inaccurate.
Counting on the fairness ensures we can reliably predict the statistical likelihood of outcomes, and it is a crucial assumption when solving problems involving dice rolls. Hence, when the exercise states 'a fair die,' we are assured of equal chances for all faces, which is fundamental to determining accurate probabilities.