91Ó°ÊÓ

Two faces of a six-sided die are painted red, two are painted blue, and two are painted yellow. The die is rolled three times, and the colors that appear face up on the first, second, and third rolls are recorded. a. Let \(B B R\) denote the outcome where the color appearing face up on the first and second rolls is blue and the color appearing face up on the third roll is red. Because there are as many faces of one color as of any other, the outcomes of this experiment are equally likely. List all 27 possible outcomes. b. Consider the event that all three rolls produce different colors. One outcome in this event is \(R B Y\) and another \(R Y B\). List all outcomes in the event. What is the probability of the event? c. Consider the event that two of the colors that appear face up are the same. One outcome in this event is \(R R B\) and another is \(R B R\). List all outcomes in the event. What is the probability of the event?

Step by step solution

01

List all 27 possible outcomes of the experiment

To achieve this, we must consider all possible combinations of the colors on the die when rolled three times. These combinations can be represented as follows: 1. RRR 2. RRB 3. RRY 4. RBR 5. RBB 6. RBY 7. RYR 8. RYB 9. RYY 10. BRR 11. BRB 12. BRY 13. BBR 14. BBB 15. BBY 16. BYR 17. BYB 18. BYY 19. YRR 20. YRB 21. YRY 22. YBR 23. YBB 24. YBY 25. YYR 26. YYB 27. YYY

02

List all outcomes of event A (all three rolls produce different colors)

We need to find all the outcomes where all three colors are different. The outcomes are: 1. RBY 2. RYB 3. BRB 4. BYR 5. YBR 6. YRB

03

Calculate the probability of event A

The probability of event A is the number of outcomes in event A divided by the total number of outcomes. So, the probability of event A is: P(A) = \(\frac{6}{27}\) = \(\frac{2}{9}\)

04

List all outcomes of event B (two of the colors that appear face up are the same)

We need to find all the outcomes where exactly two colors are the same. The outcomes are: 1. RRR 2. RRB 3. RYR 4. BRR 5. BBR 6. BYB 7. YYY 8. YRY 9. YBY 10. RBR 11. RYB 12. RBB 13. BYR 14. YBR 15. YYR 16. RYR 17. YRB 18. BYY

05

Calculate the probability of event B

The probability of event B is the number of outcomes in event B divided by the total number of outcomes. So, the probability of event B is: P(B) = \(\frac{18}{27}\) = \(\frac{2}{3}\) The probabilities for the required events are \(\frac{2}{9}\) for all three rolls producing different colors and \(\frac{2}{3}\) for two of the colors appearing face up being the same.

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Key Concepts

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

Probability of Independent Events

Understanding the probability of independent events is essential in discrete mathematics, particularly when working with scenarios involving multiple stages or trials. Independent events are those whose outcomes do not affect one another. For instance, rolling a die multiple times can be seen as a series of independent events because the result of one roll does not influence the outcome of the subsequent rolls. To calculate the probability of multiple independent events occurring in sequence, the individual probabilities are multiplied together. This is because the chance of each event is not altered by the occurrence of the others. For example, if you wanted to find the probability of rolling a blue face and then a red face on our painted die, you would multiply the individual probabilities of each event happening consecutively.

Sample Space Enumeration

Sample space enumeration refers to the listing of all possible outcomes of an experiment. It forms the foundation for calculating the probability of events by providing a clear picture of the entire set of outcomes that could occur. In the given die-rolling exercise, the sample space includes 27 possible outcomes, each representing a sequence of colors that could appear after rolling the die three times. By enumerating all possible outcomes we can categorize events, like all three rolls producing different colors, and calculate their probabilities with accuracy. Enumerating the sample space also helps in visualizing complex probability scenarios and ensures that we encapsulate all eventualities.

Combinatorics in Probability

Combinatorics is a field of mathematics concerning the counting, arrangement, and combination of elements within a set. In probability, combinatorics comes into play prominently when determining the number of possible outcomes in an event. The exercise provides an elegant example of how combinatorial principles can be utilized to list outcomes involving rolling a die with different colored faces. By using the basic combinatorial rules, you determine that with three rolls and three possible colors, there are a total of 27 outcomes (3 colors raised to the power of 3 rolls). This method of counting is vital in calculating probabilities and becomes increasingly useful as the complexity of the scenarios increases, such as in card games or when dealing with several dice or numerous trials.

Equally Likely Outcomes

When an experiment's outcomes are equally likely, each outcome has the same chance of occurring as any other. This is a fundamental concept in calculating probabilities, especially in fair games or symmetrical objects like our six-sided die. In the situation where the die is unbiased and each color has an equal number of faces, the outcomes of consecutive rolls are equally likely. This simplifies the process of probability calculation because it allows us to assume that the likelihood of landing on any given color is the same. To find the probability of an event, we can divide the number of favorable outcomes (where the event occurs) by the total number of equally likely outcomes in the sample space. This leads to the quick calculation of probabilities, as demonstrated in the exercise, and is a practical assumption in many real-world applications.