Question

What is the probability of throwing a 3, 5 or an even number with a 6-sided dice?

Short answer

The probability is \(\frac{5}{6}\).

Step-by-step solution

- Understanding the Dice

A standard 6-sided die has the numbers 1, 2, 3, 4, 5, and 6. Each face of the die has an equal probability of appearing when the die is rolled.

- Listing Favorable Outcomes

We need to count how many of these numbers satisfy the condition of being either a 3, a 5, or an even number. The even numbers on the die are 2, 4, and 6. So, the numbers that qualify are: 2, 3, 4, 5, 6.

- Counting Total Outcomes

Since the die has 6 sides, there are 6 possible outcomes for each roll of the die.

- Counting Favorable Outcomes

The favorable outcomes are: 2, 3, 4, 5, 6. Therefore, there are 5 favorable outcomes.

- Calculating Probability

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Thus, the probability is: \[ P = \frac{5}{6} \]

Key concepts

Probability Calculation

Probability is essentially the measure of the likelihood of an event happening. To find the probability of an event, you divide the number of favorable outcomes by the total number of possible outcomes. For example, in the exercise, you needed to find the probability of rolling a 3, 5, or an even number on a 6-sided die.
To do this:

• Identify the favorable outcomes (2, 3, 4, 5, 6).
• Count the total number of possible outcomes (6 since there are 6 faces on a die).
• Calculate the probability: \[ P = \frac{5}{6} \]It’s a simple yet powerful way to predict how likely an event is to occur!

Dice Outcomes

When you roll a standard 6-sided die, each side has an equal chance of landing face up. This means there are exactly 6 possible outcomes: 1, 2, 3, 4, 5, and 6. Each roll is completely independent of the previous roll, meaning the previous outcome does not affect the next one.
If the question is about specific numbers (like 3, 5, or even numbers), identifying these outcomes helps in counting how many ways those specific numbers can appear. In our example:

• The numbers are 1, 2, 3, 4, 5, and 6.
• For our exercise, favorable outcomes include the numbers 2, 3, 4, 5, and 6.
Each number has a \[ \frac{1}{6} \] chance of appearing.

Favorable Outcomes

Favorable outcomes are those which fulfill the conditions of the event we are interested in. For our exercise, we are looking for outcomes that are 3, 5, or even numbers (2, 4, 6).

• Firstly, list all possible outcomes: 1, 2, 3, 4, 5, 6.
• Then filter the ones that meet the requirements: 2, 3, 4, 5, and 6.
• This gives us 5 favorable outcomes.

Identifying these quickly and accurately is crucial in solving probability problems. The higher the number of favorable outcomes compared to the possible outcomes, the higher the probability of the event happening.

Basic Combinatorics

Combinatorics is the area of mathematics dealing with combinations. It helps us determine how many different ways we can combine items. In the context of a 6-sided die, each face is an independent event.
One basic principle in combinatorics is the counting principle, which tells us how to count the number of possible outcomes. For a die:

• Each roll has 6 possible outcomes (1 to 6).
• When multiple conditions are present (like in our exercise), it’s about counting how many outcomes satisfy those conditions.

In the given problem, we identify numbers that are 3, 5, or even (2, 4, 6), giving us 5 favorable outcomes. This straightforward combinatorial counting allows us to solve the probability equation accurately.