Problem 13 - Algebra and Trigonometry

Question

A die is rolled. Find the probability of getting $$\text {an odd number.}$$

Step-by-Step Solution

Answer

The probability of rolling an odd number on a die is 0.5

Step 1 - Identify total number of outcomes

A standard die has 6 faces, each with a different number from 1 to 6, so the total number of outcomes is 6 when a die is rolled.

Step 2 - Identify favorable outcomes

Since we are looking for an odd number, the favorable outcomes are the odd numbers on the die -- 1, 3, and 5. So there are 3 favorable outcomes.

Step 3 - Apply Probability Formula

The probability of an event is calculated as the ratio of the number of favorable outcomes to the total number of outcomes. So, the probability of getting an odd number is \( \frac{3}{6} = 0.5 \)

Key Concepts

Outcomes of a Die Roll

When we talk about outcomes in probability, we're referring to the possible results that can happen from an experiment, like rolling a die. A standard six-sided die is a cube with each face displaying a unique number of dots ranging from 1 to 6. Each roll of the die produces one of these six numbers, and since the die has six faces, there are exactly six possible outcomes. These outcomes are all equally likely if the die is fair, meaning each number has an equal chance of appearing on top after a roll.

Understanding the concept of outcomes is crucial for solving any probability problem, as it forms the foundation of knowing how probable any given event is. For instance, when rolling a die, one could roll a 1, a 2, a 3, a 4, a 5, or a 6. Each number is a result of the die roll, and hence each one represent an outcome.

Favorable Outcomes

In the context of probability, 'favorable outcomes' refer to the outcomes which satisfy the condition of the event we're interested in. They are the 'successes' that we're counting when we're calculating probability. For example, if we want to find the probability of rolling an odd number with a six-sided die, we first need to identify which numbers on the die are odd. In this case, the odd numbers are 1, 3, and 5. These are the 'favorable outcomes' because they are the numbers that meet our criteria for success (being odd in this scenario).

In other probability exercises, always ensure to correctly identify the favorable outcomes for the event being questioned. This will avoid any errors in your calculation and help you understand the true likelihood of the event.

Probability Formula

The probability formula is a mathematical way to express how likely an event is to happen. It is the ratio of the number of favorable outcomes to the total number of possible outcomes. This ratio gives us a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain. The formula is written as:
\[ P(Event) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

For example, as we discussed with the die roll, to find the probability of getting an odd number, we take the number of favorable outcomes (3) and divide it by the total number of outcomes (6). This gives us a probability of \( \frac{3}{6} \), which simplifies to 0.5 or 50%. Thus, there is a fifty percent chance of rolling an odd number on a six-sided die.

Remembering and correctly applying the probability formula is essential for solving any probability-related problem accurately.