TU BCA Probability and Statistics Master Guide: Binomial, Poisson, Normal Distribution & Hypothesis Testing (Solved Numericals)
Author: Bhuban Subedi | Subject: Probability and Statistics (CAST202) | Semester: Third Semester
In computer science and modern artificial intelligence, statistical models drive machine learning classification, network traffic modeling, queueing theory, and data analytics. In the Tribhuvan University BCA third semester, Probability and Statistics (CAST202) tests students on foundational probability, theoretical distributions, and inferential hypothesis testing.
In the 60-mark external TU board examination, numerical calculations represent over 70% of the paper. Scoring high requires mastering the mathematical formulas for Binomial, Poisson, and Normal distributions, alongside calculating $Z$-scores and $p$-values.
In this guide, I will solve standard TU board numericals step-by-step.
1. Conditional Probability & Bayes’ Theorem
For two events $A$ and $B$ where $P(B) > 0$:
$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$
Bayes’ Theorem Formula:
$$P(A_i|B) = \frac{P(A_i) \cdot P(B|A_i)}{\sum_{k=1}^n P(A_k) \cdot P(B|A_k)}$$
2. Discrete Probability Distributions
+-------------------+-------------------------------------------+-----------------------+
| Distribution | Probability Mass Function (PMF) | Mean & Variance |
+-------------------+-------------------------------------------+-----------------------+
| **Binomial** | $$P(X = r) = \binom{n}{r} p^r q^{n-r}$$ | $$\mu = np$$ |
| *(Fixed trials n)*| where $q = 1 - p$ | $$\sigma^2 = npq$$ |
+-------------------+-------------------------------------------+-----------------------+
| **Poisson** | $$P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!}$$ | $$\mu = \lambda$$ |
| *(Rare events)* | where $\lambda = np$ | $$\sigma^2 = \lambda$$|
+-------------------+-------------------------------------------+-----------------------+
Solved TU Board Problem 1: Binomial Distribution
Problem: A computer network transmits packets where the probability of a defective packet is $p = 0.1$. If a random sample of $n = 6$ packets is inspected, calculate the probability that:
1. Exactly 2 packets are defective.
2. At least 1 packet is defective.
Solution:
Here, $n = 6, \; p = 0.1, \; q = 1 – p = 0.9$.
Part 1: Exactly 2 Defective ($r = 2$):
$$P(X = 2) = \binom{6}{2} (0.1)^2 (0.9)^{6-2}$$
$$\binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15$$
$$P(X = 2) = 15 \times (0.01) \times (0.9)^4 = 15 \times 0.01 \times 0.6561 = \mathbf{0.0984 \; (9.84\%)}$$
Part 2: At Least 1 Defective ($P(X \ge 1)$):
Using the complement rule:
$$P(X \ge 1) = 1 – P(X = 0)$$
$$P(X = 0) = \binom{6}{0} (0.1)^0 (0.9)^6 = 1 \times 1 \times 0.531441 = 0.531441$$
$$P(X \ge 1) = 1 – 0.531441 = \mathbf{0.4686 \; (46.86\%)}$$
Solved TU Board Problem 2: Poisson Distribution
Problem: The average number of server crashes per week on a university web portal is $\lambda = 2$. Find the probability that in a randomly selected week:
1. No crashes occur ($r = 0$).
2. Exactly 3 crashes occur ($r = 3$). (Given: $e^{-2} = 0.1353$)
Solution:
Using the Poisson formula $P(X = r) = \frac{e^{-\lambda} \lambda^r}{r!}$:
Part 1: No Crashes ($r = 0$):
$$P(X = 0) = \frac{e^{-2} \cdot 2^0}{0!} = \frac{0.1353 \times 1}{1} = \mathbf{0.1353 \; (13.53\%)}$$
Part 2: Exactly 3 Crashes ($r = 3$):
$$P(X = 3) = \frac{e^{-2} \cdot 2^3}{3!} = \frac{0.1353 \times 8}{6} = \frac{1.0824}{6} = \mathbf{0.1804 \; (18.04\%)}$$
3. Continuous Normal Distribution & Standard Z-Scores
The Standard Normal Variable ($Z$) standardizes any continuous normal variable $X \sim N(\mu, \sigma^2)$:
$$Z = \frac{X – \mu}{\sigma}$$
+-------------------------------------------------------------------------------+
| NORMAL DISTRIBUTION EMPIRICAL RULE |
| * Area within \mu +/- 1\sigma = 68.27% of total data. |
| * Area within \mu +/- 2\sigma = 95.45% of total data. |
| * Area within \mu +/- 3\sigma = 99.73% of total data. |
+-------------------------------------------------------------------------------+
4. Hypothesis Testing: The 5-Step Statistical Procedure
+---+-------------------------------+---------------------------------------------------------------+
| # | Step | Description |
+---+-------------------------------+---------------------------------------------------------------+
| 1 | **Formulate Hypotheses** | $H_0$ (Null Hypothesis: $\mu = \mu_0$) versus |
| | | $H_1$ (Alternative Hypothesis: $\mu \neq \mu_0$ or $> / <$) |
+---+-------------------------------+---------------------------------------------------------------+
| 2 | **Set Significance Level** | Common $\alpha = 0.05 \; (5\%)$ or $\alpha = 0.01 \; (1\%)$. |
+---+-------------------------------+---------------------------------------------------------------+
| 3 | **Select Test Statistic** | Use **Z-Test** if $n \ge 30$ or population $\sigma$ is known. |
| | | Use **t-Test** (Student's $t$) if $n < 30$ and $\sigma$ unknown.|
+---+-------------------------------+---------------------------------------------------------------+
| 4 | **Determine Critical Value** | For two-tailed Z-test at $\alpha = 0.05 \implies Z_{crit} = 1.96$.|
+---+-------------------------------+---------------------------------------------------------------+
| 5 | **Decision Rule** | If $|Z_{cal}| > Z_{crit}$, **Reject $H_0$**; otherwise accept.|
+---+-------------------------------+---------------------------------------------------------------+
Frequently Asked Questions (FAQ)
Q1: What is the difference between Type I and Type II errors?
A Type I Error ($\alpha$) occurs when a true null hypothesis is incorrectly rejected. A Type II Error ($\beta$) occurs when a false null hypothesis fails to be rejected.
Q2: When is the Poisson distribution used as an approximation to the Binomial?
When the number of trials $n$ is very large ($n \to \infty$) and the probability of success $p$ is very small ($p \to 0$), such that $\lambda = np$ remains a constant finite number.
