Solved Exercise Q:8.18 to 8.24 (Part#3) By Sher Muhammad Chaudhary |Chapter#8

Introduction

This tutorial provides a step-by-step guide on solving exercises from Chapter 8, focusing on discrete probability distributions, including the binomial distribution. It is designed for students and professionals looking to enhance their understanding of probability concepts and their applications.

Step 1: Understand the Binomial Distribution

• The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials (yes/no experiments).
• Key parameters:
  • n: number of trials
  • p: probability of success on each trial
• Important formulas:
  • Probability Mass Function (PMF): [ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]
  • Mean: ( np )
  • Variance: ( np(1-p) )

Step 2: Identify the Problem

• Read through the exercise carefully to understand what is being asked.
• Note the values of n (trials), p (probability of success), and k (number of successes).
• Example problem: "What is the probability of getting exactly 3 successes in 5 trials with a success probability of 0.6?"

Step 3: Calculate the Probability

• Plug the values into the PMF formula.
• For the example:
  • n = 5, k = 3, p = 0.6
• Calculation:
  • Calculate (\binom{5}{3}): [ \binom{5}{3} = \frac{5!}{3!(5-3)!} = 10 ]
  • Then compute: [ P(X = 3) = 10 \times (0.6)^3 \times (0.4)^{2} ]
  • Complete the calculation to find the probability.

Step 4: Interpret the Results

• Analyze the calculated probability in the context of the problem.
• Discuss what the result means in practical terms, such as likelihood of the event occurring.

Step 5: Review Common Pitfalls

• Ensure correct identification of n, p, and k.
• Double-check calculations for factorials and powers.
• Remember that the sum of all probabilities in a binomial distribution equals 1.

Conclusion

In this tutorial, we explored the binomial distribution, identified the problem parameters, calculated probabilities using the PMF formula, and interpreted the results. This systematic approach can be applied to various problems involving discrete probability distributions. For further practice, consider reviewing additional exercises and applications of other distributions such as hypergeometric or Poisson distributions.