Problem 16 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to: [Jan. 12,2019 (I)] (a) \(\frac{200}{6^{5}}\) (b) \(\frac{150}{6^{5}}\) (c) \(\frac{225}{6^{5}}\) (d) \(\frac{175}{6^{5}}\)

Step-by-Step Solution

Answer

The probability is \( \frac{200}{6^5} \) (option a).

Step 1: Define the Experiment's Conditions

We need to find the probability of rolling two consecutive 4s on a fair six-sided die, occurring exactly on the fifth roll. For this sequence to occur precisely on the fifth roll, the specifics of the first four rolls are significant.

Step 2: Sequence Analysis for First Four Rolls

For the two consecutive 4s to occur on the 4th and 5th rolls, the first three rolls must not result in a consecutive 4. This means that the sequence, up to the 3rd roll, can be any combination that ends without a '4', or one that ends with a single '4'. Effectively, we cover three cases: no '4', or just one '4' (anywhere in the sequence).

Step 3: Calculating Sequences Without Succeeding '4s'

For the first three rolls ending without '44', you can have:- 'NNN': No '4's at all.- 'NN4': Ending on a single '4'.- 'N4N': '4' only second position.Here, 'N' represents any number that is not '4'.Thus, each position choice (non-'4') has a probability of \(\frac{5}{6}\) and a '4' has \(\frac{1}{6}\).

Step 4: Total Probability of Acceptable Sequences for First Four Rolls

Number of suitable sequences:- 'NNN': \( (\frac{5}{6})^3 \)- 'NN4', 'N4N', '4NN': \( 3 \times (\frac{5}{6})^2 \times \frac{1}{6} \)Calculate total for acceptable sequences:\( (\frac{5}{6})^3 + 3(\frac{5}{6})^2(\frac{1}{6}) \).

Step 5: Calculating Probability for Last Two Rolls

To complete the experiment on the 5th roll, the 4th roll must be '4' and the 5th roll must also be '4'. The probability for each is \(\frac{1}{6}\) and \(\frac{1}{6}\) respectively.

Step 6: Calculate Total Probability

Multiply the probability of the acceptable sequences of the first four rolls by the probability of rolling '4' in the last two:\[ ( (\frac{5}{6})^3 + 3(\frac{5}{6})^2 \times (\frac{1}{6}) ) \times \frac{1}{6} \times \frac{1}{6} \].Simplify the result based on calculations and options available.

Step 7: Final Probability Computation and Simplification

Calculate the total probability as follows:- Contribution of 'NNN': \( (\frac{5}{6})^3 \times \frac{1}{36} \)- Contribution for 'NN4', 'N4N': \( 3 \times (\frac{25}{216}) \times \frac{1}{36} \)Calculate both contributions:\[ (\frac{125}{1296}) + 3(\frac{25}{1296}) \,- \frac{75}{1296} = \frac{200}{1296} = \frac{200}{6^5} \]. Thus, the answer is \( \frac{200}{6^5} \).

Key Concepts

Random Experiment

A random experiment is an action or process that leads to one or more possible outcomes, where the result cannot be predicted with certainty before it occurs. The concept of randomness is crucial because it allows us to assign probabilities to different outcomes in a systematic way.
For example, when rolling a die, we consider it a random experiment since each roll can result in any number from 1 to 6, and the result varies randomly from one trial to the next. This unpredictability is what makes it random.

• Each trial of the experiment is independent. This means that the outcome of one roll doesn’t affect subsequent rolls.
• Every outcome has a probability attached, allowing prediction over the long term.

Understanding random experiments is essential when dealing with probabilities because they set the stage for applying probability theory to analyze and predict outcomes.

Fair Die

A fair die is an unbiased six-sided die, where each face (1 through 6) has an equal chance of landing face up. This means the probability of rolling any one of these numbers is \(\frac{1}{6}\).
The fairness of a die is fundamental when discussing probability, as it ensures that each roll is random and not influenced by any external factors.

• A die is considered fair if it is perfect in shape and density, with each side having an equal likelihood of landing face up.
• In probability, assuming a fair die ensures calculations are based on uniform distribution.

This concept is crucial for exercises involving probability with dice because it allows us to apply straightforward calculations without needing to adjust for bias.

Consecutive Rolls

Consecutive rolls refer to rolling a die multiple times in succession, where we are interested in the order and results of each roll.
In probability exercises like this, we often focus on specific sequences that occur across these consecutive rolls. For example, if you roll a die five times, you might be looking for a specific sequence like "4, 4" appearing in rolls four and five.

• Common focus areas include finding sequences, comparisons between rolls, and patterns that may occur during the rolls.
• Understanding the dependency between specific rolls is crucial, especially when targeting particular outcomes.

In this exercise, we're particularly interested in an event—two 4s appearing consecutively—which is why understanding consecutive rolls is necessary.

Mathematical Probability

Mathematical probability is a branch of mathematics that deals with calculating the likelihood of various outcomes of random events. It gives us a way of quantifying how likely an outcome is provided all outcomes are equally probable.
In mathematics, probability values range from 0 to 1, where 0 indicates an impossible event and 1 indicates a certainty.

• Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
• In our exercise, we perform multiple calculations involving different sequences to arrive at the likelihood of a specific event—achieving two consecutive 4s on the fifth roll.

Probability is foundational in determining the occurrences of sequences in random experiments like dice rolling and plays an integral part in understanding events' likelihood and certainty.