Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

If a single die is tossed, find the probability of obtaining an odd number or a prime number.

Step-by-Step Solution

Answer

The probability is \( \frac{2}{3} \).

Step 1: Understanding the problem

We need to find the probability of obtaining a number that is either odd or prime when a single die is tossed. A single die has six faces with numbers 1, 2, 3, 4, 5, and 6.

Step 2: Identify outcomes for odd numbers

Odd numbers on a die are 1, 3, and 5. Therefore, there are 3 outcomes (1, 3, 5) that are odd.

Step 3: Identify outcomes for prime numbers

Prime numbers on a die are 2, 3, and 5. Therefore, there are 3 outcomes (2, 3, 5) that are prime.

Step 4: Combine outcomes for odd and prime

List out all numbers that are either odd or prime from the die. The numbers are 1, 2, 3, and 5. Note that 3 and 5 are both odd and prime, resulting in a total of 4 unique favorable outcomes.

Step 5: Calculate the probability

The probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. The total number of possible outcomes when tossing a die is 6. Thus, probability \( P \) is \( \frac{4}{6} \). Simplify to get \( \frac{2}{3} \).

Key Concepts

Odd Numbers

An odd number is any number that cannot be evenly divided by two. When you divide an odd number by 2, there will always be a remainder. Odd numbers are a fundamental part of mathematics. In the context of a die, which has six faces numbered from 1 to 6, the odd numbers you can roll are 1, 3, and 5.

• Odd numbers increase by adding 2 to the previous odd number (e.g., 1, 3, 5, etc.).
• They form half of the natural numbers starting from 1.
• In a standard six-faced die, there are exactly 3 odd numbers.

Understanding odd numbers helps know what numbers appear during games involving dice, such as board games.

Prime Numbers

Prime numbers are unique and fascinating. These numbers have only two distinct positive divisors: 1 and themselves. They are a crucial concept in number theory and mathematics because they are the building blocks of whole numbers. On a standard die:

• The prime numbers are 2, 3, and 5.
• Each of these numbers can only be divided evenly without a remainder by 1 and themselves.
• There are no other prime numbers between 1 and 6.

In many problems like this one on dice, knowing which numbers are prime helps identify potential outcomes when rolling the die.

Die Tossing

Die tossing is a simple yet fascinating probabilistic event. When you toss a die, there are 6 possible outcomes because each face is equally likely to be facing up. This means when you roll a standard die once, every number from 1 to 6 has an equal probability of occurring:

• Each outcome is equally likely to occur.
• The probability of any one specific number appearing is therefore 1 in 6, or \( \frac{1}{6} \).
• Tossing a die is a common example of a discrete uniform distribution.

This concept underpins the randomness of die rolling in board games and other applications, showing the equal chances every face holds.

Favorable Outcomes

Favorable outcomes are the specific outcomes we are interested in when calculating probabilities. In the context of probability, an outcome is favorable if it meets the conditions of the event we're examining. For the task of finding an odd or prime number when throwing a die:

• The favorable outcomes are 1, 2, 3, and 5.
• These are numbers that meet either criteria (odd or prime).
• Each additional number meeting both criteria (like 3 and 5, which are both odd and prime) does not change the total unique outcomes.
• To find the probability, divide the number of favorable outcomes (4) by the total number of outcomes (6), resulting in \( \frac{4}{6} = \frac{2}{3} \).

Understanding favorable outcomes is vital for accurately calculating probabilities in probabilistic events.