Problem 12 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A die is tossed. The event an even or a prime number occurs on the top of the die is (a) \(\\{2,5\\}\) (c) \(\\{1,2,3,5\\}\) (b) \(\\{2,3,4,5,6\\}\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is (b) \(\{2,3,4,5,6\}\).

Step 1: Identify Even Numbers

A die has six faces with numbers 1 to 6. The even numbers are 2, 4, and 6.

Step 2: Identify Prime Numbers

The prime numbers on a die (numbers 1 to 6) are 2, 3, and 5. A prime number is a number greater than 1 that has no divisors other than 1 and itself.

Step 3: Combine Even and Prime Numbers

Combine the even numbers {2, 4, 6} and the prime numbers {2, 3, 5} to form the set of numbers for the event. This set is {2, 3, 4, 5, 6}.

Step 4: Match with Given Choices

Compare the combined set {2, 3, 4, 5, 6} with the given options. Option (b) \(\{2,3,4,5,6\}\) matches the set.

Key Concepts

Even Numbers

When we talk about even numbers, we're referring to numbers that are divisible by 2. This means that if you divide an even number by 2, you will get a whole number without any remainder. In the context of a dice roll, which has faces numbered from 1 to 6, the even numbers are 2, 4, and 6.

Identifying even numbers is pretty straightforward. To check if a number is even, you can divide it by 2. If the result is an integer (a whole number), then it's even. Understanding even numbers is crucial in probability, especially when dealing with equally divided sets, such as when we're interested in half the outcomes of a particular event.

• Stable and predictable patterns due to their divisibility by 2.
• On a standard six-sided die, even numbers account for exactly half of the numbers.

Prime Numbers

Prime numbers are a little special in mathematics. A prime number is a number greater than 1 that cannot be formed by multiplying two smaller natural numbers. In simpler terms, it has exactly two distinct positive divisors: 1 and the number itself. In the context of dice with faces numbered from 1 to 6, the prime numbers are 2, 3, and 5.

It’s helpful to recognize prime numbers quickly, as they often show up in different areas of math and probability. Remember, the prime numbers within our standard die are essential for problems requiring combinations or unions, such as the overlap with even numbers.

• They're not as frequent as even numbers but crucial for calculations.
• Each number on the die between 1 and 6 has only itself and 1 as factors.

Die Toss

Rolling a die is a classic example in probability and statistics, often used due to its uniformity — each side of a fair, six-sided die has a 1 in 6 chance of facing up. When discussing an event involving a die toss, it's useful to consider both the outcome's likelihood and the nature of the numbers themselves.

In exercises involving dice, you might calculate the probability of specific outcomes, like rolling an even number or a prime number. For example, if you're asked the probability of rolling an even number or a prime number, you should first identify these numbers and then consider their union - essentially which numbers fulfill one or both criteria.

• There are 6 possible outcomes: 1, 2, 3, 4, 5, 6.
• A common exercise asks for the union of different criteria in these outcomes.