Problem 59 - A Complete Resource Book in Mathematics for JEE Main

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 probability 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

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

Step 1: Define the Event A

We need to find the probability of the event where the minimum face value rolled is not less than 2 and the maximum face value is not greater than 5. Let's call this event \(A\).

Step 2: Calculate Total Outcomes

Since the die is rolled four times, each roll having 6 outcomes, the total number of possible outcomes is \(6^4\). Therefore, the total number of outcomes is \(6 \times 6 \times 6 \times 6 = 1296\).

Step 3: Calculate Outcomes for Event A

For the event \(A\), the possible face values range from 2 to 5. Hence, there are 4 possible face values for each die roll (2, 3, 4, or 5). The number of outcomes where all rolls result in values between 2 and 5 is \(4^4\). Thus, this is \(4 \times 4 \times 4 \times 4 = 256\).

Step 4: Compute Probability for Event A

The probability of event \(A\) occurring is the ratio of the favorable outcomes (from Step 3) to the total number of possible outcomes (from Step 2). Thus, the probability is \(\frac{256}{1296}\).

Step 5: Simplify the Probability

Simplify \(\frac{256}{1296}\) by dividing the numerator and the denominator by their greatest common divisor, which is 16. The simplified probability is \(\frac{16}{81}\).

Key Concepts

Unbiased Die

An unbiased die is a fair six-sided die where each face has an equal chance of landing face-up. It means that the probability of rolling any of the numbers from 1 to 6 is the same, which is \(\frac{1}{6}\). It is crucial in probability exercises as it ensures that all outcomes are equally likely, providing a standard basis for calculations. You don't need to worry about imbalanced results due to favoritism of any number. Here are some important characteristics:

• Each face is equally probable when rolled.
• The total of probabilities for all faces equals 1.
• It is usually used in theoretical probability exercises.

In this problem, since the die is rolled four times, every roll having an equal chance creates a framework where calculations rely on uniform probabilities.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally distinct and not continuous. Unlike calculus or algebra, it deals with countable, separate values often useful in computer science, logic, and combinatorics. In contexts involving games of chance, such as dice, discrete mathematics helps in analyzing outcomes that are distinct events. For instance, each die roll is a discrete event with clear boundaries: a number between 1 to 6 is rolled with no in-between. Some essential aspects include:

• Finite state spaces: Limited number of events or outcomes.
• Logical reasoning and counting: Dependence on structured logic to determine probabilities.
• Combinatorics: Applying principles to count possible outcomes or configurations.

When rolling dice, each outcome is distinct and countable, providing a typical example of discrete mathematics at work.

Probability Theory

Probability theory is the mathematical framework used for calculating undetermined events or quantities. It helps us predict the likelihood of potential outcomes. It is especially useful when dealing with random events, such as rolling a die. Probability theory was used to calculate the event (let's call it "Event A") in our problem, where we desired a minimum face value of 2 or more, and a maximum face value of 5 or less. Some key concepts include:

• Random Experiments: Each die roll is an experiment with unknown outcomes.
• Event Space: All the possible outcomes, like rolling between 1 and 6.
• Probability Measure: Calculating likelihood using ratios of favorable to total outcomes.
• Expected Outcomes: Using probability to anticipate results over many trials.

In the example, probability theory was harnessed to find the likelihood of the event happening, which involves counting how many rolls from 4 produce numbers only between 2 and 5.

Combinatorics

Combinatorics is an area in mathematics focused on counting, arrangement, and combination of elements within a set. It plays a pivotal role in probability and helps in solving problems like determining how many ways there are to arrange or choose items. When using combinatorics in probability with dice, one often counts the number of ways a desired outcome can occur. In the exercise given, combinatorics were utilized to find how many combinations exist where each die roll result is between 2 and 5. Key combinatorial concepts include:

• Permutations: Arranging a set where order matters.
• Combinations: Choosing items where order does not matter.
• Binomial Coefficients: Often used in probability distributions.

In the problem, combinatorics was essential for determining 4 raised to the 4th power, representing the number of possible outcomes when rolling dice such that rolls fit within given constraints.