Question

Suppose that sample space \(\Omega_{1}\) is chosen to model the experiment of rolling a pair of dice and that the probability density function \(p\) assigned to \(\Omega_{1}\) is \(p(\omega)=1 / 36\) for \(\omega \in \Omega_{1}\). Under these assumptions, compute the probability of rolling a sum of 3. Compare your answer to the answers of \(1 / 18\) and \(1 / 11\) obtained in the text, and discuss.

Short answer

The probability of rolling a sum of 3 is 1/18, matching one of the given answers.

Step-by-step solution

Understanding the Problem

We are given the sample space \( \Omega_{1} \) for rolling two dice and that each outcome has a probability \( p(\omega) = \frac{1}{36} \). We need to calculate the probability of the event where the sum of the two dice rolls equals 3.

Identifying Favorable Outcomes

For the two dice to sum to 3, the possible outcomes are (1,2) and (2,1). These are the only combinations out of two dice that result in a sum of 3. Thus, there are 2 favorable outcomes.

Computing Probability of Sum 3

With each outcome having a probability of \( \frac{1}{36} \), the probability of obtaining a sum of 3 is the sum of the probabilities of these favorable outcomes: \[P(\text{sum of 3}) = 2 \times \frac{1}{36} = \frac{2}{36} = \frac{1}{18}.\]

Comparison with Given Answers

The calculated probability \( \frac{1}{18} \) matches one of the given answers. Comparing it to the other answers, we find it does not match \( \frac{1}{11} \), confirming the correct probability is indeed \( \frac{1}{18} \).

Key concepts

Sample Space

Probability theory often begins with the concept of a sample space, denoted as \( \Omega \). This is the set of all possible outcomes of a random experiment. For example, when rolling two six-sided dice, the sample space \( \Omega_1 \) includes all ordered pairs from \( (1,1) \) to \( (6,6) \). Each pair represents a potential outcome, capturing all combinations of numbers that could appear on the dice.

The size of the sample space is important for calculating probabilities. With two dice, there are 36 possible outcomes in \( \Omega_1 \). These outcomes are equally likely if the dice are fair, making each probability \( \frac{1}{36} \) under a uniform distribution.

Understanding the sample space helps identify favorable outcomes, which are the specific outcomes we are interested in. In the case of rolling a sum of 3, the favorable outcomes are \((1,2)\) and \((2,1)\). Thus, knowing and analyzing the sample space allows us to calculate probabilities for specific events.

Probability Density Function

A probability density function (PDF) is a function that assigns probabilities to the outcomes in a sample space. For discrete sample spaces, the PDF is often denoted by \( p \). Each outcome is paired with a probability, which, in the discrete case, must sum up to 1 over all possible outcomes.

In our dice example, we use the PDF \( p(\omega) = \frac{1}{36} \) for each outcome \( \omega \in \Omega_1 \). This expresses the likelihood of each pair of dice rolls occurring, acknowledging that all outcomes are equally probable with fair dice. The uniform probability distribution reflects this equal likelihood.

The PDF enables us to compute the probability of more complex events by summing the probabilities of their constituent outcomes. Here, calculating the probability of a sum of 3 involves summing the probabilities of the outcomes \((1,2)\) and \((2,1)\), yielding \( \frac{2}{36} \). Therefore, the PDF is a crucial tool for quantifying uncertainty in a consistent mathematical framework.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, which contrasts with continuous mathematics where numbers can vary smoothly. In probability, discrete mathematics applies when our sample space comprises distinct, countable outcomes.

Dice rolling is a prime example of a discrete mathematical problem. Every roll results in one pair from a finite set of combinations \((1,1), (1,2), ..., (6,6)\). These outcomes are discrete because there is no value between them; you either roll \((1,2)\) or you do not. There is no rolling, say, \((1.5, 2.5)\).

Discrete probability theory is equipped to handle these scenarios. It uses the set theory to define events and establish probabilities. Calculating the probability of discrete events, like rolling a sum of 3 with two dice, involves counting favorable outcomes and applying the mathematics of finite sets. Understanding discrete mathematics is essential for solving problems where outcomes are distinct and individually identifiable.