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Chapter 4: Problem 108

Suppose that you are performing the probability experiment of rolling one fair six-sided die. Let F be the event of rolling a four or a five. You are interested in how many times you need to roll the die in order to obtain the first four or five as the outcome. 鈥� \(p\) = probability of success (event F occurs) 鈥� \(q \)= probability of failure (event F does not occur) a. Write the description of the random variable X. b. What are the values that \(X\) can take on? c. Find the values of \(p\) and \(q\). d. Find the probability that the first occurrence of event F (rolling a four or five) is on the second trial.

Short Answer

Expert verified
X is the number of rolls until the first 4 or 5; values are positive integers. \(p = \frac{1}{3}\), \(q = \frac{2}{3}\). Probability of first success on second trial is \(\frac{2}{9}\).

Step by step solution

01

Describe the Random Variable X

Let the random variable \( X \) represent the number of trials required to obtain the first four or five when rolling a fair six-sided die. In other words, \( X \) is the number of rolls needed until a success (rolling a 4 or 5) occurs for the first time.
02

Determine the Possible Values of X

Since \( X \) represents the number of rolls needed to achieve the first success, \( X \) can take on any positive integer value starting from 1, i.e., \( X = 1, 2, 3, \ldots \) etc.
03

Calculate the Probability of Success (p)

The probability \( p \) is the chance of rolling a 4 or a 5 in a single roll of a six-sided die. Since there are 2 favorable outcomes (4 or 5) out of 6 equally likely possibilities, \( p = \frac{2}{6} = \frac{1}{3} \).
04

Calculate the Probability of Failure (q)

The probability \( q \) is the chance of NOT rolling a 4 or a 5 in a single roll, which means rolling a 1, 2, 3, or 6. There are 4 out of 6 such outcomes, so \( q = \frac{4}{6} = \frac{2}{3} \).
05

Calculate Probability of First Success on Second Trial

To find the probability that the first occurrence of event F happens on the second trial, we need the probability of failing on the first trial and succeeding on the second. This probability is \( q \cdot p = \frac{2}{3} \times \frac{1}{3} = \frac{2}{9} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variable
In probability theory, a random variable is a fundamental concept that serves as a function assigning numerical values to the outcomes of a random process. They are used to quantify the outcomes of a stochastic event in terms of number.
In our given exercise, the random variable is denoted as \( X \). More specifically, \( X \) represents the number of trials, which in this case is the number of times we roll a six-sided die, needed to achieve a specific event. Here, the event is rolling a four or a five.
  • This variable is termed as a 'discrete' random variable since it can take a distinct set of values.
  • The possible values of \( X \) are all positive integers (i.e., \( 1, 2, 3, \ldots \)), as you must roll the die at least once to succeed.
Working with random variables allows us to model and analyze the randomness inherent in processes, which is essential for predicting probabilities and outcomes.
Geometric Distribution
The geometric distribution is a key concept when dealing with probability experiments where the goal is to count the number of trials needed for the first success. It is considered the simplest discrete probability distribution.
For our exercise's random variable \( X \), the number of trials to get the first four or a five follows a geometric distribution. This arises from the independent nature of each roll and the constant probability of success.
  • The probability of success (\( p \)) when rolling a die is \( \frac{1}{3} \), as there are 2 successful outcomes (rolling a 4 or 5) out of 6 possible outcomes.
  • Conversely, the probability of failure (\( q \)) is \( \frac{2}{3} \), indicating outcomes where neither a four nor a five is rolled.
The probability that the first success happens on a specific trial (e.g., the second trial) can be computed using \( q^{n-1} \times p \), where \( n \) is the desired trial number.
Independent Trials
The concept of independent trials is crucial in probability theory, especially when analyzing repeated experiments. Independent trials imply that the outcome of one trial does not affect the outcome of another.
When rolling a six-sided die, each roll is independent, meaning whether a 4 or 5 comes up does not influence the result of subsequent rolls. This makes the probability \( p = \frac{1}{3} \) for success and \( q = \frac{2}{3} \) for failure valid for each roll.
  • This independence is what allows us to use the formula for geometric distribution: \( P(X=n) = q^{n-1} \cdot p \).
  • It justifies the calculation of rolling a successful number on a specific trial, as seen with calculating the probability of success on the second trial being \( \frac{2}{9} \).
Understanding independent trials helps in accurately modelling real-world scenarios where multiple independent events occur, ensuring that the probability measures remain consistent across observations.

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