Problem 24 - Algebra 2

Question

A standard number cube is tossed. Find each probability. \(P(4 \text { or less than } 6)\)

Step-by-Step Solution

Answer

The probability of landing a '4 or less than 6' is \(\frac{5}{6}\).

Step 1: Identify Possible Outcomes

A standard cube has six faces, each bearing a different number from 1 to 6. Therefore, the total number of outcomes when a cube is tossed is six - {1, 2, 3, 4, 5, 6}.

Step 2: Identify Favorable Outcomes

The event is to land a '4 or less than 6'. The numbers that satisfy this requirement are {1, 2, 3, 4, 5}. So, there are five favorable outcomes.

Step 3: Calculate the Probability

Probability of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes. So, the probability \(P\) is \(P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{5}{6}\).

Key Concepts

Number Cube

A number cube, commonly known as a die, is a small object with six square faces. Each face is marked with a unique number from 1 to 6. Tossing a number cube is a fundamental probability experiment, widely used to explore the basics of probability.

Every time the cube is tossed, it can land on one of its six faces. This makes it an ideal tool for learning basic probability concepts. Understanding how a number cube works and analyzing the possible results it can yield helps lay the groundwork for comprehending more complex probability scenarios.

Outcomes

In probability, an outcome is a possible result of a chance experiment, like tossing a number cube. When you toss a standard number cube, there are six potential outcomes, corresponding to each of the six faces of the cube. These outcomes can be represented as:

• 1
• 2
• 3
• 4
• 5
• 6

Each outcome is equally likely when tossing a fair number cube. Understanding that each number represents an outcome is essential when calculating probability. By identifying all outcomes, you can better assess which results are favorable for a specific event.

Favorable Outcomes

Favorable outcomes are the specific outcomes that satisfy the conditions of the event you are considering. For example, in the exercise, the "event" is defined as rolling a number "4 or less than 6" when a number cube is tossed. The favorable outcomes in this case are all the numbers that satisfy this condition.

These favorable outcomes would be:

• 1
• 2
• 3
• 4
• 5

Identifying the favorable outcomes is crucial because it directly influences how you calculate probability. By clearly determining which numbers are favorable, calculating the probability becomes a straightforward process that involves counting and comparing.

Event Probability

The probability of an event is a measure of the likelihood that the event will occur. The key to calculating event probability is understanding the ratio of favorable outcomes to total possible outcomes. With our number cube example, once you've identified both the total number of possible outcomes (6) and the number of favorable outcomes (5), calculating the probability becomes simple.

The probability formula is: \[P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\]
Applying this formula to our example gives: \[P(4 \text{ or less than 6}) = \frac{5}{6}\]
This fraction represents how likely it is to roll a number "4 or less than 6" when tossing the cube. Probability values range from 0 to 1, where 0 means the event is impossible and 1 means it is certain. In this case, \(\frac{5}{6}\) indicates a highly probable event, reflecting that most of the numbers on the cube meet the condition.