Problem 63 - Calculus for Biology and Medicine

Question

A random experiment consists of rolling a fair die until the first time an even number appears. Find the probability that the first even number appears on the third trial.

Step-by-Step Solution

Answer

The probability is \(\frac{1}{8}\).

Step 1: Understand the Problem

We are rolling a fair die until we get an even number, and we want the first even number to show on the third trial. We need to calculate this probability.

Step 2: Analyze Possible Outcomes for Each Roll

A fair die has even numbers: 2, 4, 6, and odd numbers: 1, 3, 5. The probability of rolling an odd number is therefore \(\frac{3}{6} = \frac{1}{2}\), and the probability of rolling an even number is \(\frac{3}{6} = \frac{1}{2}\).

Step 3: Identify Needed Combination of Rolls

For the first even number to appear on the third trial, the first two rolls must yield odd numbers, followed by an even number on the third roll. The sequence is odd, odd, even.

Step 4: Calculate the Probability of the Sequence

The probability of getting odd on the first trial is \(\frac{1}{2}\), odd on the second trial is \(\frac{1}{2}\), and even on the third trial is \(\frac{1}{2}\). Thus, the probability of the sequence is: \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}\).

Step 5: Final Answer

The probability that the first even number appears on the third trial is \(\frac{1}{8}\).

Key Concepts

Random Experiment

In the context of probability theory, a random experiment is any process or action that yields a set of possible outcomes. Each time you perform this experiment, the outcome can vary, and it is impossible to predict the result with certainty.
For example, rolling a fair six-sided die is a classic random experiment. Each roll is independent, and each time you roll, there are six potential outcomes: the numbers 1 through 6.
In the exercise, the experiment is defined by rolling a die until achieving a result—a specific type of number, even in our case. This unpredictability is a core feature of random experiments, as each outcome is governed by chance, not determinism.

• Outcome: Result of the experiment (like rolling a 4).
• Sample Space: The set of all possible outcomes (like {1, 2, 3, 4, 5, 6} for a die).
• Event: A possible subset of outcomes (such as rolling an even number).

Even and Odd Numbers

When working with dice or similar contexts, understanding the distinction between even and odd numbers is crucial.
The fundamental principle here is simple: when a number is divided by 2, if the remainder is zero, it's even. If the remainder is one, it's odd.
Using a six-sided die as an example:

• Even numbers: 2, 4, 6 (as they are all divisible by 2 with no remainder).
• Odd numbers: 1, 3, 5 (as they each leave a remainder of 1 when divided by 2).

The exercise requires you to track sequences of such numbers when rolling a die, which influences the calculation of probabilities. Recognizing these numbers helps in grouping possible outcomes of random experiments.

Probability Calculation

Probability calculation is key in predicting the likelihood of different outcomes in a random experiment.
It's a measure quantifying how likely events are to occur, expressed as a fraction, a decimal, or a percentage.
In probability theory, the formula is given as:\[ P(Event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]In the dice example, if you want to calculate the probability of rolling an even number, you have three possible successful outcomes (2, 4, 6) out of six total outcomes, thus:\[ P( ext{Even}) = \frac{3}{6} = \frac{1}{2} \]Similarly, for rolling an odd number, it's also \( \frac{1}{2} \).
In the exercise, to find the probability of obtaining an even number on the third roll after two odd numbers, we multiply the probabilities of each step in the sequence. Hence:\[ P( ext{Odd, Odd, Even}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

Sequences

A sequence in probability refers to an ordered list of events. The outcome of each event in the sequence can affect the next, especially if dependent events are considered. However, with independent events - like dice rolls - each event does not impact others' outcomes.
Consider a sequence of rolling a die, where each roll represents a single event:

• The order in which events occur matters—e.g., Odd, Odd, Even is not the same as Even, Odd, Odd.
• In the exercise, the sequence we're interested in is Odd, Odd, Even.
• Identifying the sequence requires understanding possible outcomes at each step.

Each roll being independent ensures that past rolls do not influence future probability calculations, allowing the use of established probabilities for each event in a sequence.