Problem 58 - Algebra 2

Question

A standard number cube is tossed. Find each probability. \(P(\text { even or } 7)\)

Step-by-Step Solution

Answer

Since getting 7 on a 6-sided dice is impossible, we only consider the probability of getting an even number, which is \(\frac{1}{2}\).

Step 1: Identify Favorable Outcomes

In this case, the favorable outcomes are either the dice landing on an even number. The even numbers on a 6-sided dice are 2, 4 and 6. So there are 3 favorable outcomes.

Step 2: Identify Total Outcomes

On tossing a standard number cube or a dice, the total possible outcomes are numbers 1 to 6. So, there are 6 possible outcomes.

Step 3: Calculate Probability

The formula for calculating probability is number of favorable outcomes divided by total number of outcomes. In this case, the number of favorable outcomes is 3 (either 2, 4 or 6) and the total number of outcomes is 6. So, \(P(\text { even })\) = \(\frac{3}{6} = \frac{1}{2}\).

Key Concepts

Favorable Outcomes

When dealing with probability, the concept of favorable outcomes is essential. These are the outcomes of an event that we are interested in. For example, if we're tossing a die and looking to get an even number, the even numbers are our favorable outcomes.
In simpler terms, favorable outcomes are the results we want. They provide us meaningful insight into calculating probabilities effectively. Knowing the number of favorable outcomes is the first step in determining the probability of an event occurring.

On a standard six-sided die, the numbers 2, 4, and 6 are even numbers. Therefore, if we want to find the probability of rolling an even number, we identify these numbers as our favorable outcomes. In this scenario, there are 3 favorable outcomes.

Dice Probability

Rolling a six-sided die is a classic example of probability. Dice probability revolves around understanding how likely it is to land on a specific number or group of numbers when a die is tossed.
A standard die has six faces, each showing numbers from 1 to 6. When you roll the die, each side has an equal chance of landing face up. Therefore, each side represents one possible outcome in our probability calculations.

• Total outcomes on a standard die: 6 (one for each face).
• When rolling a die, each number from 1 to 6 appears with an identical probability of \(\frac{1}{6}\).

Understanding that a die provides an equal chance for each of its faces helps in determining the odds of any event, such as rolling an even number or any other specified criteria.

Even Numbers

Even numbers are integers that can be divided by 2 without leaving a remainder. They hold a special place, especially when calculating probability with dice, due to their distinct characteristics.
When you roll a standard six-sided die, the even numbers you might roll are 2, 4, and 6. These numbers are divisible by 2 and form half of the possible outcomes when considering a die roll. In probability, identifying these numbers is crucial when asked for the likelihood of rolling an even number.

• On a standard die: the even numbers are 2, 4, and 6.
• These numbers are undivided by 2 and are commonly targeted in probability questions involving dice.

This understanding aids us in efficient calculation by narrowing down the outcomes that meet the criteria of being even, which is essential for accurate probability computations.

Calculating Probability

Calculating probability involves comparing the number of favorable outcomes to the total possible outcomes. It's a way to determine the likelihood of an event. The basic formula for probability is:\[P( ext{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\]In the case of rolling an even number with a six-sided die, the steps are simple:

• Identify the favorable outcomes: For even numbers, it's 2, 4, and 6. Hence, there are 3 favorable outcomes.
• Determine the total outcomes: A standard die has 6 sides, so 6 total outcomes.
• Apply the probability formula: \(P(\text{even}) = \frac{3}{6} = \frac{1}{2}\)

This means that when you roll your die, there is a 50% chance of landing on an even number. By consistently applying this formula, probability calculations become straightforward and intuitive.