Question

Find the probability that Event \(\mathrm{A}\), drawing a spade on a single draw from a deck of cards, and Event \(\mathrm{B}\), rolling a total of 7 on a single roll of a pair of dice, will both occur.

Short answer

The probability that both Event A (drawing a spade from a deck of cards) and Event B (rolling a total of 7 on a single roll of a pair of dice) will occur is \(1/24\).

Step-by-step solution

Find the probability of drawing a spade from a deck of cards (Event A)

A standard deck of cards has 52 cards, with 13 cards belonging to each of the four suits (hearts, diamonds, clubs, and spades). To find the probability of drawing a spade, we will divide the number of spades in the deck by the total number of cards in the deck: P(A) = Number of spades / Total number of cards P(A) = 13 / 52 P(A) = 1/4

Find the probability of rolling a total of 7 on a single roll of a pair of dice (Event B)

There are a total of 36 possible outcomes when rolling a pair of dice, as each die has 6 sides and each side can be paired with any side on the other die. We will find the number of ways to get a total of 7 on the dice and then divide this by the total number of possible outcomes: P(B) = Number of ways to get a total of 7 / Total number of possible outcomes There are six ways to get a total of 7 using two dice: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). P(B) = 6 / 36 P(B) = 1/6

Find the probability of both events occurring (Event A and Event B)

As both events (A and B) are independent, we can now find their joint probability by multiplying their individual probabilities: P(A and B) = P(A) * P(B) P(A and B) = (1/4) * (1/6) P(A and B) = 1/24 Therefore, the probability that both events A and B will occur is 1/24.

Key concepts

Probability Theory

Probability theory is a branch of mathematics focusing on the analysis of random events. The likelihood of an event occurring is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Within this framework, the calculation of the probability of independent events, such as drawing a spade from a deck of cards and rolling a total of 7 on a pair of dice, involves understanding that the outcome of one event does not affect the outcome of another. When events A and B are independent, the probability of both A and B occurring is the product of their individual probabilities, mathematically represented as: \[\begin{equation} P(A \text{ and } B) = P(A) \times P(B) \end{equation}\]
This fundamental concept is a cornerstone of probability theory and is pivotal in determining the outcome likelihood in scenarios ranging from simple games to complex stochastic processes.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, arrangement, and combination of elements within a set according to certain rules.

In the context of probability, combinatorics helps us quantify the number of possible outcomes and discern the particular events of interest. For instance, when analyzing dice rolls, combinatorics tells us that with a pair of six-sided dice, there are 36 distinct possible outcomes.

To calculate the probability of rolling a total of 7, we need to identify all combinations that sum up to 7. These combinations are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), leading to six favorable outcomes. By knowing the total and favorable outcomes, we can use combinatorics to determine that the probability of this event is \[\begin{equation} \frac{6}{36} \end{equation}\]
which simplifies to \[\begin{equation} \frac{1}{6} \end{equation}\].
Understanding the combinatorial principles behind these calculations is essential for solving more complex probability questions.

Discrete Mathematics

Discrete mathematics encompasses a variety of topics that involve discrete elements and applies to fields such as computer science and information theory. One of the key areas within discrete mathematics is the study of finite or countable structures, unlike continuous systems which are the focus of calculus.

Probability problems, like the one involving a deck of cards and dice, fall within the realm of discrete mathematics, because they involve counting specific outcomes from a finite set of possibilities. In our exercise, the focus is on discrete probability – the study of probability when dealing with countable outcomes. The deck consists of a finite number of cards, and the dice have a limited number of sides, leading to a limited set of possible rolls.

Understanding discrete probability helps students analyze situations where outcomes are distinct and finite, and lays the foundation for fields like computer science where discrete structures and algorithms play a crucial role in problem-solving.