Problem 11 - Algebra 2

Question

Two standard number cubes are tossed. State whether the events are mutually exclusive. Explain your reasoning. The numbers are equal; the sum is odd.

Step-by-Step Solution

Answer

Yes, the events are mutually exclusive. If the numbers rolled are equal, their sum is always even and hence the sum cannot be odd. Similarly, if the sum is odd, the numbers rolled cannot be equal. Thus one event excludes the other from occurring simultaneously.

Step 1: Understanding The Events

There are two events in question: the numbers on two dice are equal, and the sum of the numbers is odd. From the possible outcomes of a dice roll {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}, it's observed that every time the numbers are equal, their sum is an even number. Therefore, if the numbers are equal, it's impossible to get an odd sum.

Step 2: Analyzing The Second Event

The other possible outcome could be that the sum of the numbers rolled is odd. For such a situation, the outcomes are {(1,2), (2,1), (1,4), (4,1), (1,6), (6,1), (3,2), (2,3), (3,4), (4,3), (3,6), (6,3), (5,2), (2,5), (5,4), (4,5) or (5,6),(6,5)}. In this case, it's seen that none of the numbers rolled on both dice are the same. Hence, if the sum is odd, the numbers rolled cannot be equal.

Step 3: Conclusion

Given the analysis in step 1 and 2, where if one event occurs, the other cannot; it is clear that these two events are mutually exclusive.

Key Concepts

Number Cubes

Number cubes are commonly known as dice. Each standard number cube has six faces numbered from 1 to 6. When we roll a number cube, we can get any one of these numbers, and each face has an equal chance of landing face up.
This means there are six possible outcomes for rolling one die. When two number cubes are rolled together, as in our exercise, the situation becomes intriguing.

• Each die acts independently of the other, meaning that the result of one roll does not affect the other.
• There are a total of 6 × 6 = 36 possible outcomes when two dice are rolled.
• These outcomes can be listed as ordered pairs: (1,1), (1,2), ..., (6,6).

Understanding the behavior of two number cubes is crucial for solving problems related to them, like finding the probability of certain outcomes.

Probability

Probability is a measure of how likely an event is to occur. It is expressed as a fraction or a decimal between 0 and 1. A probability of 0 means the event can never happen, while a probability of 1 means the event is certain to happen.
When dealing with number cubes, we often calculate the probability of certain outcomes when they are rolled together.

• To find the probability of an event, we count the number of successful outcomes and divide by the total number of possible outcomes.
• For two dice, this means considering all 36 possible combinations of rolls.
• If there is only one out of the 36 combinations that meets our criteria, its probability is 1/36.

In our exercise, the events of having the two dice show equal numbers and having a sum that is odd must be considered. Given how these events are defined, they lead directly into discussions about mutually exclusive events.

Dice Outcomes

Dice outcomes are the different results you can achieve when number cubes are rolled. With two dice, there are several interesting observations to make about their outcomes.

• Outcomes where both dice show the same number are like (1,1), (2,2), ... up to (6,6). These always result in even sums, like 2, 4, ..., 12.
• Outcomes giving an odd sum involve one die landing on an even number and the other on an odd number, such as (1,2), (2,1), (3,4), (4,3), etc.
• To determine if two events are mutually exclusive, we need to establish if they can happen at the same time.

In our case, since outcomes where the numbers are equal always produce even sums, and outcomes where the sum is odd never have equal numbers, these two are mutually exclusive. This means when one event happens, there's no chance for the other to occur simultaneously.