Problem 20 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

An unbiased die with faces marked \(1,2,3,4,5\) and 6 is rolled four times. Out of four face values obtained, the probabiity that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is, (A) \(\frac{16}{81}\) (B) \(\frac{1}{81}\) (C) \(\frac{80}{81}\) (D) \(\frac{65}{81}\)

Step-by-Step Solution

Answer

Option (A) \( \frac{16}{81} \).

Step 1: Determine the Total Number of Outcomes

When rolling a 6-sided die four times, each die roll can land on any of the 6 different outcomes. Therefore, the total number of outcomes from rolling the die four times is calculated as: \[ 6^4 = 1296 \]

Step 2: Define the Desired Outcomes

We want the minimum face value to be not less than 2 and the maximum face value to be not greater than 5. This means all numbers on each roll must be between 2 and 5, inclusive. Thus, the possible values for each roll are 2, 3, 4, or 5.

Step 3: Determine the Number of Desired Outcomes

Since each die can result in one of four values (2, 3, 4, or 5) and there are four rolls, the number of desired outcomes is: \[ 4^4 = 256 \]

Step 4: Calculate Probability

The probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5 is given by the ratio of the number of successful outcomes to the total number of outcomes:\[ P = \frac{256}{1296} = \frac{16}{81} \]

Step 5: Compare and Conclude

The calculated probability \( \frac{16}{81} \) matches option (A). Therefore, the probability is \( \frac{16}{81} \).

Key Concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics that deals with discrete elements that use distinct values. It is essentially about understanding countable objects. In contrast to continuous mathematics, which deals with continuous data like curves and functions, discrete mathematics focuses on distinct and separate objects.

In our problem of rolling a die, each roll represents a discrete event. Every result is distinct (from 1 to 6), and the outcomes can be approached through counting techniques. This is where discrete mathematics shines, as it leverages principles of counting, permutations, combinations, and probability in situations where every outcome is individually observed and can be computed precisely.

By analyzing problems using discrete mathematics, we can break down complex scenarios into simpler components. For example, determining the number of outcomes when rolling a die multiple times involves straightforward multiplication. This structured approach helps us to systematically solve probabilities and understand complex events.

Dice Rolling

Dice rolling is a common example used to demonstrate concepts in probability and chance. When you roll a fair six-sided die, you are engaging in an experiment that has a set number of possible outcomes, each of which is equally likely.

In our exercise, a die is rolled four times. Each roll results in a new value, ranging from 1 to 6. However, our exercise focuses on the situation where we only consider values from 2 to 5. This focus limits the number of outcomes, making the problem a bit easier to handle.

When you roll the die four times, you're creating a series of independent events:

• Each roll has an equal probability for each of the possible outcomes.
• The result of one roll does not affect the others.

This independence is crucial in calculating probability because it allows us to use multiplication for finding the total number of possible outcomes, as was done with the formula \(6^4\) or \(4^4\). Understanding this helps us grasp why certain calculations in probability theory work the way they do.

Probability Theory

Probability theory is a mathematical framework for quantifying uncertain events or phenomena. In this context, it allows us to calculate and foresee the likelihood of different outcomes from a set of defined possibilities. The central idea is to establish the probability of an event occurring by comparing the number of favorable outcomes to the total number of possible outcomes.

Probability is often given as a fraction, representing the number of successful or favorable outcomes over the total outcomes. In our dice-rolling problem, we employ this by comparing how many configurations of dice rolls meet the criteria to the total number of possible configurations. The formula used here is\[ P = \frac{256}{1296} = \frac{16}{81} \]This fraction expresses the probability of rolling a series of four dice such that no number is less than 2 and none greater than 5.

The use of probability theory also involves understanding events' independence and calculating accurately using permutations and combinations when required. It helps us not only in predicting and analyzing dice games but also in fields of science, economics, engineering, and many more where risk and uncertainty are inherent.