Question

Find the number of possible outcomes in the sample space. Then list the possible outcomes. A bag contains four red cards numbered 1 through 4 , four white cards numbered 1 through 4 , and four black cards numbered 1 through 4 . You choose a card at random.

Short answer

The number of possible outcomes in this sample space is 12. The possible outcomes are: Red 1, Red 2, Red 3, Red 4, White 1, White 2, White 3, White 4, Black 1, Black 2, Black 3, Black 4.

Step-by-step solution

Identifying the Types of Cards and Numbers

Initially we need to identify the types of cards and the numbers allotted to them. There are three types of cards: red, white and black. Each card has four numbers on them, which are 1, 2, 3, 4.

Calculate the Sum of All Possible Outcomes

In probability, the sum of all possible outcomes forms the sample space. We can calculate the sample space by multiplying the number of card types by the number of numbers, which is 3 types of cards times 4 numbers, adding up to 12 possible outcomes.

Identify All Possible Outcomes

Each card picked is an outcome. For the red cards, the possible outcomes are Red 1, Red 2, Red 3, and Red 4. The same applies for the white and black cards, leading to the outcomes White 1, White 2, White 3, White 4, Black 1, Black 2, Black 3, Black 4.

Key concepts

Probability Outcomes

When it comes to understanding probability, one of the first concepts to grasp is that of probability outcomes. Probability outcomes are the possible results of an experiment or a random trial. For instance, if you flip a coin, there are two possible outcomes: heads or tails. In the context of our exercise involving the selection of a card from a bag of differently colored and numbered cards, each individual card represents a unique outcome.

More specifically, in our exercise, we calculate the number of outcomes by considering all the possible cards that could be chosen. With each color—red, white, and black—having 4 numbered cards, we end up with 12 distinct outcomes when a single card is drawn. These outcomes form the sample space, which is essential for calculating probabilities as it represents all the possible results of the experiment. By listing each card as a separate outcome (e.g., Red 1, White 2), we create a complete view of the sample space, which is foundational for further probabilitistic analysis.

Permutations and Combinations

Diving deeper into the world of probability will inevitably bring you to the concepts of permutations and combinations, which are methods for counting the number of ways we can arrange or select items. A permutation counts how many different ways we can arrange a set of items where the order matters. Conversely, a combination counts the number of ways to select items from a set where the order is not important.

In the context of our card example, if we were instead asked to draw two cards without replacement, we'd need to use combinatorial calculations. Permutations would apply if the order of drawing mattered (e.g., drawing a red card first and then a white card is different from drawing a white card first and then a red card), while combinations would be used if the order did not matter. Understanding when to use permutations or combinations is crucial for solving more complex probability problems.

Basic Probability Principles

Now, let's get into the essence of basic probability principles, which govern how we calculate the likelihood of a particular event. The foundational principle of probability is that the probability of an event is equal to the number of favorable outcomes divided by the total number of possible outcomes in the sample space. This principle is expressed mathematically as:

\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
Applying this to our exercise, if we wanted to know the probability of drawing a red card, we'd see that there are 4 favorable outcomes (Red 1, Red 2, Red 3, Red 4) out of a total of 12 outcomes in the sample space. So, the probability would be \(\frac{4}{12}\) or \(\frac{1}{3}\). Grasping these basic principles is essential for all future work in probability and helps students reason through problems logically, providing them with a firm foundation for more advanced concepts in probability and statistics.