Problem 20 - 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 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

The probability is \( \frac{16}{81} \) which corresponds to option (A).

Step 1: Identify the Range for the Desired Outcome

We need to find the probability that the minimum face value when rolling the die four times is not less than 2, and the maximum face value is not greater than 5. This implies that all the die outcomes must be between 2 and 5 inclusive.

Step 2: Calculate Total Possible Outcomes

When an unbiased die is rolled 4 times, total possible outcomes are calculated by multiplying the number of outcomes for each roll. Hence, we have:\[ 6^4 = 1296 \]

Step 3: Calculate Favorable Outcomes

For the favorable event, the die only shows numbers from 2 to 5. The number of favorable outcomes on a single roll is 4 (i.e., 2, 3, 4, 5). Therefore, the total number of favorable outcomes when rolled 4 times is:\[ 4^4 = 256 \]

Step 4: Compute Probability of Favorable Events

The probability of the favorable event occurring is the ratio of favorable outcomes to total possible outcomes:\[ \frac{256}{1296} = \frac{16}{81} \]

Step 5: Select the Correct Option

Comparing the computed probability with the given options, we see that option (A) \( \frac{16}{81} \) matches our result.

Key Concepts

Combinatorics

Combinatorics is a fascinating area of mathematics focused on counting, arranging, and grouping different objects. When rolling a die multiple times, combinatorics helps us calculate the total number of possible outcomes. In our example with a die being rolled four times, each die roll has 6 potential outcomes. To find all combinations over four rolls, we multiply these possibilities together, resulting in a total outcome count of \( 6^4 = 1296 \). Understanding combinatorics allows students to visualize how many different sequences can be formed from multiple random events, like rolling a die.

Discrete Mathematics

Discrete Mathematics is the study of mathematical structures that are fundamentally discrete, as opposed to continuous. It provides the foundation for understanding probabilities in structured scenarios, like rolling a die. In discrete math, events like die rolls are considered distinct and finite, allowing us to use combinatorial methods to solve problems. When considering the outcomes of a die roll, discrete math helps us evaluate each possible result as a separate, countable object. This allows for structured calculation of probabilities, such as determining that 256 possible outcomes meet specific conditions out of a total of 1296 possible results.

Random Variables

Random variables are a key concept in probability theory and statistics, representing the numerical outcomes of random phenomena. In the context of our die rolling problem, the results of each die roll are random variables, ranging from 1 to 6. By defining our random variables clearly, we can apply mathematical methods to predict outcomes and analyze their probabilities. For instance, by restricting our random variables to values between 2 and 5, the probabilities are altered, giving us a particular subset of all possible results on the die. This kind of refinement allows us to calculate the probability of specific events with precision, such as finding that the minimum value is not less than 2 and the maximum is not greater than 5, resulting in a probability of \( \frac{16}{81} \).