Problem 18 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The probability of getting a number greater than 2 in throwing a die is (a) \(1 / 3\) (b) \(2 / 3\) (c) \(1 / 2\) (d) \(1 / 6\)

Step-by-Step Solution

Answer

The probability of getting a number greater than 2 is \( \frac{2}{3} \) (option b).

Step 1: Understand the Die

A standard die has 6 faces, numbered from 1 to 6. We need to find the probability of rolling a number greater than 2.

Step 2: Identify Favorable Outcomes

The numbers greater than 2 on a die are 3, 4, 5, and 6. Therefore, there are 4 favorable outcomes (3, 4, 5, 6).

Step 3: Total Possible Outcomes

Since a standard die has 6 faces, there are a total of 6 possible outcomes when rolling the die.

Step 4: Calculate the Probability

The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Therefore, the probability of rolling a number greater than 2 is \( \frac{4}{6} \).

Step 5: Simplify the Probability Fraction

Simplify \( \frac{4}{6} \) to \( \frac{2}{3} \) by dividing both the numerator and the denominator by 2.

Key Concepts

Favorable Outcomes

When calculating probability, it's important to first identify the favorable outcomes of the event you're interested in. For example, when rolling a six-sided die, we want to determine the number of outcomes that result in a number greater than 2.
For a standard dice roll, the numbers are 1 through 6. However, only 3, 4, 5, and 6 are greater than 2. Thus, we say there are 4 favorable outcomes because these are the outcomes that satisfy our condition.
Identifying favorable outcomes correctly leads to accurate probability calculations. Always take a moment to list out these possibilities before proceeding.

Total Possible Outcomes

The concept of total possible outcomes refers to all the outcomes that can occur in a given situation. In the case of rolling a standard die, there are 6 possible outcomes because there are 6 faces on the die, numbered 1 through 6.
Total possible outcomes serve as the denominator in the probability fraction. Understanding this helps clarify why we divide the number of favorable outcomes by the total number of possible outcomes to calculate probability.

• This consideration is crucial because it sets the basis for calculating probability in a manner where the likelihood of all possible events is represented equally.

Once you know the total possible outcomes, the next step is to determine how many of these outcomes are favorable, which we've discussed earlier.

Simplifying Fractions

Once you have your probability fraction, it's essential to simplify it to its lowest terms to make it easier to comprehend. In our example, the fraction derived was \(\frac{4}{6}\).
To simplify a fraction, divide both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).

In the fraction \(\frac{4}{6}\), the GCD of 4 and 6 is 2. By dividing both 4 and 6 by 2, you get \(\frac{2}{3}\). This is the simplest form of the fraction.
Using simplified fractions is a common standard in probability calculations because it provides a clearer picture of the likelihood of an event occurring. Always aim to reduce fractions to their simplest form for accurate and intuitive interpretations.