Problem 57 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The required probability is \(0.5\) or \(50\% \)

Step 1: Identify Total and Favorable Outcomes

A standard number cube or die has 6 faces with numbers 1 to 6. So, there are a total of 6 outcomes. The favorable outcomes for the event 'even' are 2, 4, and 6. So, there are 3 favorable outcomes.

Step 2: Calculate Probability of Even

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of outcomes. So, \(P(\text { even }) =\frac {3}{6}=0.5 \).

Step 3: Calculate Probability of 6

For the event '6', there is only one favorable outcome. So, \(P(6)= \frac{1}{6}=0.167 \).

Step 4: Calculate 'or' Probability

The probability of either event 'even' or '6' occurring is given by the sum of their individual probabilities. But since 6 is also an even number, we've counted it twice. So we need to subtract its probability once to avoid double counting. The probability is \(P(\text { even or } 6) = P(\text { even })+ P(6) - P( \text { even and 6 }) = 0.5 + 0.167 - 0.167 = 0.5 \)

Key Concepts

Number Cube

A number cube, commonly known as a die, is a cube with each of its six faces showing one of six different numbers. These numbers typically range from 1 to 6. A standard number cube is used in many games and probability exercises. It's perfect for teaching basic probability concepts because each face has an equal chance of landing face up when tossed.

Understanding a number cube is fundamental when calculating probabilities because it allows us to identify equal likelihood events. This means that when you toss a number cube, each of the numbers, from 1 to 6, has an equal probability of 1/6 because there are six faces. It's crucial to remember this uniform distribution of probability across the faces for accurate probability calculations.

Favorable Outcomes

In probability, a favorable outcome is one that matches the specific criteria or conditions described in a probability question. Identifying these outcomes is a key step in solving probability problems.

Let's look at an example with a number cube. If you want to find the probability of rolling an even number, you first identify the favorable outcomes. An even number on a standard number cube can be 2, 4, or 6. This means there are three favorable outcomes for the event 'even'.

When solving these problems, it's helpful to list possible outcomes so you clearly see which ones match your criteria. This not only helps in avoiding mistakes but also eases the calculation process.

Event Probability

Event probability is a fundamental concept that involves calculating the likelihood of a certain event occurring relative to all possible outcomes. The formula used is: \[P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}.\]

Using the example of tossing a number cube, let's calculate the probability of rolling an even number. There are 3 favorable outcomes (2, 4, 6) out of 6 possible outcomes. Therefore, the probability is:\[P( ext{Even}) = \frac{3}{6} = 0.5.\]
This calculation reflects that an even number will appear 50% of the time, under ideal conditions.

In more complex scenarios, like calculating the probability of either an event or another occurring, be mindful of overlapping outcomes. For example, finding the probability of rolling an even number or a 6 involves adding individual probabilities but subtracting the overlap. It exemplifies how to manage and correct for double-counted outcomes.