RD Sharma Class 12 Solutions Chapter 33 - Binomial Distribution (Ex 33.1) Exercise 33.1 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 33 - Binomial Distribution (Ex 33.1) Exercise 33.1
1. How do I identify if a problem from RD Sharma Class 12 Exercise 33.1 follows a Binomial Distribution?
To determine if a problem fits the Binomial Distribution model, you must check for four specific conditions which are based on Bernoulli trials. A scenario follows a binomial distribution if:
There is a fixed number of trials (n).
Each trial results in one of only two possible outcomes, typically labelled 'success' or 'failure'.
The probability of success, denoted by 'p', remains constant for every single trial.
All trials conducted are independent of each other, meaning the outcome of one does not affect the others.
If a question in Exercise 33.1 satisfies all these points, you can correctly apply the binomial formula.
2. What is the step-by-step method to calculate the probability of 'k' successes in 'n' trials for a question in RD Sharma Exercise 33.1?
To solve for the probability of a specific number of successes in Exercise 33.1, follow this precise method:
Step 1: Identify the key parameters from the problem: n (the total number of trials), p (the probability of success in a single trial), and q (the probability of failure, calculated as 1-p).
Step 2: Determine the value of k, which is the exact number of successes you need to find the probability for.
Step 3: Apply the Binomial Distribution formula: P(X = k) = C(n, k) * p^k * q^(n-k).
Step 4: Calculate the combination C(n, k) and then compute the final probability by multiplying the terms. The solutions for Exercise 33.1 clearly demonstrate this step-by-step application.
3. How are the mean and variance of a binomial distribution calculated for problems in Chapter 33?
For any binomial distribution B(n, p) described in Chapter 33, the mean and variance are found using very straightforward formulas that are essential for the CBSE Class 12 syllabus:
Mean (μ or E(X)): This represents the expected or average number of successes over many repetitions of the experiment. It is calculated as μ = np.
Variance (σ²): This measures the spread or variability of the distribution. It is calculated as σ² = npq, where q = 1-p.
Remember to always find the value of 'q' before calculating the variance.
4. What are Bernoulli trials and how are they related to the problems in Binomial Distribution Exercise 33.1?
A Bernoulli trial is a single random experiment that has exactly two possible outcomes: 'success' and 'failure'. A key feature is that the probability of success remains the same each time the trial is performed. A Binomial Distribution is a model that describes the probability of observing a certain number of successes in a fixed sequence of independent Bernoulli trials. Therefore, every question in Exercise 33.1, such as problems involving multiple coin tosses or dice rolls, is fundamentally a series of Bernoulli trials.
5. How do I solve questions in Exercise 33.1 that ask for the probability of 'at least' or 'at most' a certain number of successes?
These common problem types require calculating and summing multiple probabilities:
For 'at least k' successes (e.g., at least 3), you need to find the probabilities for k, k+1, k+2, and so on, up to n, and add them together. So, P(X ≥ k) = P(X=k) + P(X=k+1) + ... + P(X=n). A frequent shortcut is to use the complement: 1 - P(X < k).
For 'at most k' successes (e.g., at most 3), you must find the probabilities for 0, 1, 2, up to k, and add them. So, P(X ≤ k) = P(X=0) + P(X=1) + ... + P(X=k).
6. Why is it crucial to check for the independence of trials when applying the binomial distribution formula?
The independence of trials is a foundational assumption of the Binomial Distribution. The entire formula, P(X = k) = C(n, k) * p^k * q^(n-k), is derived by multiplying the probabilities of individual outcomes. According to the rules of probability, this multiplication is only valid if the events are independent. If the outcome of one trial influences the probability of success 'p' in subsequent trials (for example, drawing cards from a deck without replacement), the trials are dependent. In such a case, the binomial distribution is not the correct model to use, and applying its formula will lead to an incorrect answer.
7. How can I use the RD Sharma solutions for Exercise 33.1 to verify if my calculated mean (np) and variance (npq) are correct for a given problem?
You can use the provided solutions as a tool for verification and to deepen your understanding as per the CBSE 2025–26 curriculum. If a problem gives you the mean and variance and asks you to find 'n' and 'p', solve the system of equations (np = mean, npq = variance). Once you find your values for 'n' and 'p', substitute them back into the formulas. If your calculated μ = np and σ² = npq match the values given in the question, your method and results are correct. This self-checking process helps confirm your grasp of the concepts.
8. Can the Binomial Distribution be used if the probability of success 'p' changes with each trial? Why or why not?
No, the Binomial Distribution absolutely cannot be used if the probability of success 'p' is variable. A non-negotiable condition for a binomial experiment is that the probability of success must remain constant for all 'n' trials. If 'p' changes, the trials are no longer identical, which violates a core assumption of the model. For scenarios with changing probabilities, each step's probability must be calculated individually rather than using the simplified binomial formula.
