Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that Mary rolls a fair die until a "6" occurs. Let \(X\) denote the random variable that is the number of tosses needed for this " 6 " to occur. Find the probability distribution for \(X\) and verify that all the probabilities sum to 1 .

Step-by-Step Solution

Answer

The probability distribution is geometric with \(P(X=k) = \left(\frac{5}{6}\right)^{k-1} \cdot \frac{1}{6}\), summing to 1.

Step 1: Define the Random Variable

The random variable \(X\) represents the number of rolls needed to achieve a "6" on a fair six-sided die for the first time.

Step 2: Identify the Nature of the Distribution

The situation described fits a geometric distribution because we are repeating independent trials (rolling a die) until the first success (rolling a "6"). The probability of success \(p\) in each trial is \(\frac{1}{6}\).

Step 3: Write the Probability Mass Function

The probability mass function (PMF) of a geometric distribution for the random variable \(X\) is given by:\[ P(X = k) = (1-p)^{k-1} \cdot p \]where \(k\) is the number of trials and \(p = \frac{1}{6}\).

Step 4: Calculate the Specific Probability Expression

Substitute \(p = \frac{1}{6}\) into the PMF:\[ P(X = k) = \left(\frac{5}{6}\right)^{k-1} \cdot \frac{1}{6} \]This gives the probability for \(X\) needing exactly \(k\) rolls for the first "6".

Step 5: Verify All Probabilities Sum to 1

The sum of all probabilities for a geometric distribution should be 1:\[ \sum_{k=1}^{\infty} P(X = k) = \sum_{k=1}^{\infty} \left(\frac{5}{6}\right)^{k-1} \cdot \frac{1}{6} \]This sum is a geometric series where the first term \(a = \frac{1}{6}\) and common ratio \(r = \frac{5}{6}\), and it can be summed as:\[ \sum_{k=0}^{\infty} a r^k = \frac{a}{1-r} \]Substituting, we get:\[ \frac{\frac{1}{6}}{1-\frac{5}{6}} = 1 \]

Key Concepts

Probability Mass Function

The probability mass function (PMF) is a fundamental concept in probability theory. It provides the probabilities of different possible outcomes for a discrete random variable. In cases involving a geometric distribution, like Mary's dice rolling problem, the PMF tells us the probability of needing exactly a certain number of trials before hitting a success. The formula for the PMF of a geometric distribution is \[ P(X = k) = (1-p)^{k-1} \cdot p \]where:

• \( p \) is the probability of a success on any given trial, which in this case is \( \frac{1}{6} \) since a die has six sides.
• \( k \) is the number of trials required to achieve the first success.

This function helps in understanding how likely different outcomes are, such as how often we might expect a "6" to appear on the second, third, or tenth roll.

Random Variable

When dealing with probability, a random variable is a way to describe numerical outcomes from a random process. In Mary's scenario, the random variable \( X \) represents the outcome we're interested in, specifically, the number of times she rolls the die before getting a "6" for the first time. This highlights the power of random variables:

• They provide a bridge between abstract probability concepts and real-world situations by quantifying outcomes.
• They allow us to use mathematical tools and formulas to predict the behavior of random processes, such as rolling a die.

Random variables can be discrete or continuous. In this example, \( X \) is a discrete random variable because it can only take on countable values (1, 2, 3, etc.). Each value corresponds to different probabilities, which are captured by the probability mass function.

Geometric Series

A geometric series is a powerful mathematical tool used to sum sequences where each term is a constant multiple of the previous one. This concept is central to verifying that the probabilities in a geometric distribution add up to 1.In the example with Mary rolling a die, the series:\[ \sum_{k=1}^{\infty} \left(\frac{5}{6}\right)^{k-1} \cdot \frac{1}{6} \]is a geometric series with:

• First term \( a = \frac{1}{6} \)
• Common ratio \( r = \frac{5}{6} \)

The sum of an infinite geometric series is calculated by the formula:\[ \sum_{k=0}^{\infty} ar^k = \frac{a}{1-r} \]Applying this formula ensures that the sum of all probabilities, regardless of how many trials are needed to get that first "6", will equal 1. This sum confirms that we have accounted for all possible outcomes of the die rolls.

Probability Distribution

Probability distribution refers to a mathematical description of the likelihood of various outcomes of a random variable. It is the foundation of understanding events in probability and statistics. In Mary's dice example, the geometric distribution comes into play. This distribution models the number of trials necessary for the first success, which is landing a "6". A probability distribution ensures:

• Every possible outcome is considered, each with an associated probability calculated using the probability mass function.
• All probabilities for these outcomes sum up to 1, confirming that no possibilities have been left out.

This is pivotal for making predictions and decisions based on uncertain events, by providing a full picture of every possible scenario and their likelihood, all while ensuring our total probability remains consistent.