Hey guys! Let's dive into the solutions for Exercise 7.3 from Chapter 7 of your Class 12 Applied Maths textbook. This chapter is all about probability distributions, and Exercise 7.3 focuses specifically on the binomial distribution. Understanding this exercise is crucial for mastering the concepts in this chapter. So, grab your notebooks, and let's get started!

Understanding Binomial Distribution

Before we jump into the solutions, let’s quickly recap what binomial distribution is all about. The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. Think of it like flipping a coin multiple times and counting how many times you get heads (success). This concept is super important as it lays the groundwork for more advanced probability topics.

Key Components of Binomial Distribution

To really nail this, remember these key components:

• n: The number of trials (e.g., the number of times you flip the coin).
• p: The probability of success on a single trial (e.g., the probability of getting heads on one flip).
• q: The probability of failure on a single trial (which is simply 1 - p).
• X: The random variable representing the number of successes in n trials.

The probability mass function (PMF) for the binomial distribution is given by:

P(X = k) = C(n, k) * p^k * q^(n-k)

Where:

• P(X = k) is the probability of getting exactly k successes.
• C(n, k) is the number of combinations of n items taken k at a time, also written as "n choose k".

Make sure you're comfortable with these basics before tackling the exercise. Now, let's break down some common types of problems you'll encounter and how to solve them.

Common Problem Types in Exercise 7.3

Exercise 7.3 typically includes several types of problems related to binomial distribution. Recognizing these types will help you approach each question more effectively. So, here are a few common categories:

1. Calculating Probabilities Directly

These questions usually give you the values of n, p, and ask you to find the probability of getting a specific number of successes, like P(X = k). For example, you might be asked: "What is the probability of getting exactly 3 heads in 5 coin flips if the coin is fair?"

To solve this, just plug the values into the binomial PMF formula we discussed earlier. It's all about correctly identifying n, p, and k, and then doing the math. Accuracy is key, so double-check your calculations!

2. Calculating Cumulative Probabilities

Sometimes, you'll need to find the probability of getting at most or at least a certain number of successes. This involves calculating cumulative probabilities. For example, a question might ask: "What is the probability of getting at most 2 heads in 5 coin flips?"

Here, you need to calculate P(X ≤ 2), which means you need to find P(X = 0) + P(X = 1) + P(X = 2). Calculate each individual probability and then add them up. Remember, at most means including all values up to that number.

3. Finding Mean and Variance

Another common type involves finding the mean (expected value) and variance of a binomial distribution. The formulas for these are pretty straightforward:

• Mean (μ) = n * p
• Variance (σ²) = n * p * q

For instance, if you flip a coin 100 times, and the probability of getting heads is 0.5, the expected number of heads (mean) is 100 * 0.5 = 50. The variance would be 100 * 0.5 * 0.5 = 25.

4. Application-Based Problems

These problems put the binomial distribution into real-world scenarios. For example, "A manufacturer produces items with a 2% defect rate. In a batch of 100 items, what is the probability that exactly 3 items are defective?"

These require you to understand the context, identify the values of n and p from the problem description, and then apply the binomial PMF. Always make sure you clearly define what a “success” is in the context of the problem.

Tips and Tricks for Solving Exercise 7.3

Alright, now that we've covered the basics and the types of problems, let’s look at some tips and tricks to make solving these problems smoother. Here are some strategies that I found helpful when I was in school, and I hope they'll help you too!

1. Read the Question Carefully

This might sound obvious, but it's super important. Make sure you understand exactly what the question is asking. Pay attention to keywords like "at most," "at least," "exactly," etc. These words can completely change how you approach the problem.

2. Identify n, p, and k Correctly

The most common mistake is misidentifying these values. Take your time to correctly determine the number of trials (n), the probability of success (p), and the number of successes you're interested in (k). A little extra time here can save you from big errors later.

3. Use a Calculator

Calculating combinations and powers can be tedious and prone to errors if done manually. Use a scientific calculator to handle these calculations. Most calculators have a combination function (nCr) that can save you a lot of time.

4. Write Down the Formula

Before plugging in the values, write down the binomial PMF formula. This helps you organize your thoughts and ensures you don't miss any steps. It also makes it easier to double-check your work.

5. Check Your Answer

After you've calculated the probability, ask yourself if the answer makes sense. Probabilities should always be between 0 and 1. If you get a value outside this range, you know you've made a mistake somewhere.

6. Practice, Practice, Practice

The more problems you solve, the better you'll become at recognizing patterns and applying the correct formulas. Don't just read through the solutions; try solving the problems yourself first. This is the best way to reinforce your understanding.

Example Problems and Solutions

To really solidify your understanding, let's go through a couple of example problems together.

Example 1: Calculating Direct Probability

Problem: A fair coin is tossed 6 times. What is the probability of getting exactly 4 heads?

Solution:

1. Identify n, p, and k:

   • n = 6 (number of trials)
   • p = 0.5 (probability of getting heads on a single toss)
   • k = 4 (number of successes we want)

2. Apply the binomial PMF formula:

P(X = 4) = C(6, 4) * (0.5)^4 * (0.5)^(6-4)

3. Calculate the combination:

C(6, 4) = 6! / (4! * 2!) = (6 * 5) / (2 * 1) = 15

4. Plug in the values and calculate:

P(X = 4) = 15 * (0.5)^4 * (0.5)^2 = 15 * (0.0625) * (0.25) = 0.234375

So, the probability of getting exactly 4 heads in 6 coin tosses is approximately 0.234.

Example 2: Calculating Cumulative Probability

Problem: A die is thrown 5 times. What is the probability of getting at most 2 sixes?

Solution:

1. Identify n, p, and k:

   • n = 5 (number of trials)
   • p = 1/6 (probability of getting a six on a single throw)
   • We want to find P(X ≤ 2), which means P(X = 0) + P(X = 1) + P(X = 2)

2. Calculate the individual probabilities:

   • P(X = 0) = C(5, 0) * (1/6)^0 * (5/6)^5 = 1 * 1 * (3125/7776) ≈ 0.4019
   • P(X = 1) = C(5, 1) * (1/6)^1 * (5/6)^4 = 5 * (1/6) * (625/1296) ≈ 0.4019
   • P(X = 2) = C(5, 2) * (1/6)^2 * (5/6)^3 = 10 * (1/36) * (125/216) ≈ 0.1608

3. Add the probabilities:

P(X ≤ 2) = 0.4019 + 0.4019 + 0.1608 ≈ 0.9646

Thus, the probability of getting at most 2 sixes in 5 throws is approximately 0.9646.

Conclusion

Alright, guys, that wraps up our deep dive into Exercise 7.3 of Class 12 Applied Maths! We've covered the basics of binomial distribution, common problem types, and some handy tips and tricks to help you solve these problems effectively. Remember, the key to mastering this topic is practice. So, keep solving problems, and don't get discouraged if you make mistakes along the way. Every mistake is a learning opportunity!

Good luck with your studies, and I hope this guide helps you ace your exams! Keep grinding, and you'll get there! Happy problem-solving!