RD Sharma Class 12 Solutions Chapter 33 - Binomial Distribution (Ex 33.2) Exercise 33.2 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 33 - Binomial Distribution (Ex 33.2) Exercise 33.2
1. What is the fundamental formula used to solve problems in RD Sharma Class 12 Maths Exercise 33.2 on Binomial Distribution?
The core formula for solving problems in this exercise is the Binomial Distribution probability formula: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ. In this formula:
- P(X = r) is the probability of getting exactly 'r' successes.
- n is the total number of independent trials.
- r is the specific number of successes you are calculating the probability for.
- p is the probability of success in a single trial.
- q is the probability of failure in a single trial, which is always calculated as 1 - p.
2. How do you correctly identify the values of 'n', 'p', and 'q' from a word problem in this chapter?
To solve a problem from Exercise 33.2, you must first identify the key parameters from the question's text:
- 'n' (Number of trials): Look for the total number of times an experiment is repeated. For example, 'a coin is tossed 10 times' means n = 10.
- 'p' (Probability of success): Identify the event defined as a 'success' and find its probability for a single trial. For example, if success is 'getting a six on a die', then p = 1/6.
- 'q' (Probability of failure): Once you have 'p', the probability of failure 'q' is simply calculated as q = 1 - p. In the die example, q = 1 - 1/6 = 5/6.
3. Why is it essential for the trials in a binomial distribution to be independent?
The independence of trials is a fundamental condition for a binomial distribution because it ensures that the probability of success, 'p', remains constant for every trial. If the trials were dependent (for instance, drawing a card from a deck without replacement), the outcome of one trial would affect the probabilities of subsequent trials. This would violate the binomial model's core assumption, and a different probability method would be needed. All problems in this exercise are based on this principle of independence.
4. What is the correct method to calculate the probability of 'at least' or 'at most' a certain number of successes?
To find probabilities for a range of outcomes, you must sum the probabilities of individual outcomes:
- 'At least r' successes: This means 'r or more'. You need to calculate and add the probabilities P(X=r) + P(X=r+1) + ... + P(X=n). A common shortcut for 'at least one' is to calculate 1 - P(X=0).
- 'At most r' successes: This means 'r or fewer'. You need to calculate and add the probabilities P(X=0) + P(X=1) + ... + P(X=r).
5. What are the formulas for the mean and variance of a binomial distribution as per the CBSE Class 12 syllabus for 2025-26?
According to the CBSE syllabus, the solutions for RD Sharma Chapter 33 require knowledge of two key statistical measures:
- Mean (μ or Expected Value): The formula is μ = np. This gives the average number of successes.
- Variance (σ²): The formula is σ² = npq. This measures the spread or dispersion of the distribution.
6. How does a Bernoulli trial differ from a binomial distribution?
The terms are related but distinct. A Bernoulli trial is a single experiment with exactly two possible outcomes: success (with probability p) or failure (with probability q). For example, a single toss of a coin is a Bernoulli trial. A binomial distribution, on the other hand, describes the probability of observing a specific number of successes across a fixed number of independent and identical Bernoulli trials. Essentially, a binomial distribution is the result of performing a series of Bernoulli trials.
