Problem 116 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A person throws two dice, one the common cube and the other a regular tetrahedron, the number on the lowest face being taken in the case of tetrahedron. The probability that the sum of the numbers appearing on the dice is 6 is (A) \(\frac{1}{3}\) (B) \(\frac{1}{4}\) (C) \(\frac{1}{6}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \( \frac{1}{6} \) (Option C).

Step 1: Understand Dice Types

A common cube die has numbers from 1 to 6 on it. A regular tetrahedral die has numbers 1 to 4, one on each face. The lowest number on the tetrahedral die is equal to the number that appears on the down side after throwing.

Step 2: Determine Possible Outcomes of Each Die

The outcomes for the cube die are 1, 2, 3, 4, 5, 6. The outcomes for the tetrahedral die are 1, 2, 3, 4. We need to calculate the probability that their sum equals 6.

Step 3: Calculate Favorable Outcomes for Sum 6

List combinations where the sum of the numbers from the cube die and tetrahedral die equals 6: (2,4), (3,3), (4,2), (5,1). These are the only pairs where the sum is 6.

Step 4: Count Total Possible Outcomes

The total possible outcomes are 6 (from the cube die) multiplied by 4 (from the tetrahedral die), totaling 24 possible outcomes.

Step 5: Calculate Probability

The probability of getting a sum of 6 is the number of favorable outcomes (4) divided by the total possible outcomes (24). So, the probability is \( \frac{4}{24} = \frac{1}{6} \).

Key Concepts

Dice Probability

When working with dice, understanding their probability is a foundational skill in probability theory. Dice probability involves calculating the chances of a particular outcome when one or more dice are involved. Each side of a die represents an equally likely outcome. For example, a standard six-sided die (cube die) has six faces, numbered 1 to 6. Each face on a die has a probability of landing face up of 1/6.

A tetrahedral die, common in role-playing games, has four faces, usually numbered from 1 to 4. For each face, the probability of landing face up is 1/4. When combining these two different types of dice, as in our problem, we examine pairs of outcomes to find specific sums.

This involves listing combinations of numbers from both dice that add up to a specified sum (like 6) and calculating the probability based on possible outcomes. Understanding dice probability prepares you for more complex probability scenarios.

Combinatorics

Combinatorics is a mathematical technique for counting and arrangements. In this dice problem, it plays a key role in determining the number of potential outcomes and favorable outcomes. Combinatorics helps us figure out how many different results we can get when two or more dice are rolled.

By systematically listing and analyzing combinations, we can ensure we have considered all possibilities. For example, when seeking a sum of 6 from one cube die and one tetrahedral die, we compute combinations like (2,4), (3,3), (4,2), and (5,1). Each pair represents one possible way to achieve the desired sum.

The total number of outcomes is the product of the number of outcomes from each die. With a cube die giving 6 outcomes and a tetrahedral die giving 4, we multiply these values to find 24 possible rolling outcomes. Combinatorics ensures every outcome is accounted for precisely, which is crucial for calculating accurate probabilities.

Probability Theory

Probability theory provides the framework for predicting how likely events are to occur. It's an essential part of understanding how we approach problems like the dice problem. In any probabilistic scenario, identifying the total number of possible outcomes and the number of favorable outcomes is key.

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Here, with 4 favorable combinations resulting in a sum of 6, and a total of 24 possible outcomes from the cube and tetrahedral dice, the probability is defined as the ratio 4/24, simplifying to 1/6.

Without the laws and principles of probability theory guiding these calculations, determining probabilities from real-world scenarios like our dice problem would be far less precise. This theoretical approach helps us make informed predictions and establish clear expectations for outcomes.