Probability Theory: Events, Rules, Variables & Distributions

TAKEAWAYS:

• Probability is a measure of uncertainty defined over a sample space using three axioms (Kolmogorov), where events are subsets of outcomes and $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ for any two events.
• Conditional probability ($P(A|B)=\frac{P(A\cap B)}{P(B)}$) and Bayes’ theorem ($P(A|B)=\frac{P(B|A)P(A)}{P(B)}$) reverse probabilities to solve real-world inference problems (e.g., eSewa fraud detection).
• Random variables link numerical outcomes to probabilities: discrete (PMF) vs. continuous (PDF), with expectation $E(X)=\sum xP(X)$ or $\int xf(x)dx$ and variance $\text{Var}(X)=E(X^2)-[E(X){]}^2$.
• Key distributions include binomial (fixed trials, success/failure) and normal (bell curve, 68-95-99.7 rule), used in quality control (e.g., NTC’s call-center wait times) and finance (e.g., NEPSE stock returns).
• Independence ($P(A\cap B)=P(A)P(B)$) simplifies joint probabilities and is critical for modeling unrelated events (e.g., Daraz order delays vs. weather).
• Exam focus: Solve for unknown probabilities, derive $K$ in PMFs, compute expectations/variances, and apply Bayes’ theorem to real scenarios (e.g., Ncell’s network failure rates).

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1. Sample Spaces and Events

Probability theory starts with a sample space ($S$), the set of all possible outcomes of an experiment. An event ($A$) is any subset of $S$.

Key Definitions

• Simple event: Single outcome (e.g., rolling a 3 on a die).
• Compound event: Combination of outcomes (e.g., rolling an even number).
• Complementary event: $A^c=S\setminus A$ (e.g., not rolling a 3).

Visual: Sample Space for a Die Roll

Example 1.1: A computer virus enters via email (30%) or internet (40%). What is the probability it enters via either channel?

• Solution: $P(\text{email or internet})=P(\text{email})+P(\text{internet})-P(\text{both})$. If independent: $P(\text{both})=0.3\times 0.4=0.12$. Thus, $P=0.3+0.4-0.12=0.58$ or 58%.

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2. Probability Axioms and Rules

Probability is defined by Kolmogorov’s axioms:

1. $0\le P(A)\le 1$ for any event $A$.
2. $P(S)=1$.
3. For mutually exclusive events $A_1,A_2,\dots$, $P(A_1\cup A_2\cup \dots )=\sum P(A_i)$.

Key Rules

Example 2.1: Ncell Network Failures

• Scenario: 40% of Ncell’s calls fail due to network congestion ($N$), 30% due to hardware issues ($H$), and 15% due to both.
• Question: What is the probability a call fails due to either cause?
• Solution: $P(N\cup H)=P(N)+P(H)-P(N\cap H)=0.4+0.3-0.15=0.55$ or 55%.

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3. Conditional Probability and Bayes’ Theorem

Conditional probability measures how one event affects another.

Bayes’ Theorem

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$ Used in:

• eSewa fraud detection: If 5% of transactions are fraudulent ($F$) and 90% of frauds trigger alerts ($A$), but only 10% of non-fraudulent transactions do, what is $P(F|A)$? P(F|A) = \frac{0.9 \times 0.05}{0.9 \times 0.05 + 0.1 \times 0.95} = \frac{0.045}{0.135} \approx 0.33 \text{ (33%)}

Visual: Bayes’ Theorem Tree

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4. Random Variables

A random variable ($X$) assigns numerical values to outcomes. Two types:

1. Discrete: Finite/countable outcomes (e.g., number of heads in 10 coin flips).
2. Continuous: Uncountable outcomes (e.g., battery life in hours).

Probability Mass Function (PMF) vs. Probability Density Function (PDF)

Example 4.1: Defective Chips

A factory tests chips with PMF: $$P(X)=\{\begin{array}{ll}0.2& \text{if }X=0\\ K& \text{if }X=1\\ 0.4& \text{if }X=2\end{array}$$ Find $K$, $P(X\ge 1)$, and $E(X)$.

Solution:

1. Find $K$: $\sum P(X)=1\Rightarrow 0.2+K+0.4=1\Rightarrow K=0.4$.
2. $P(X\ge 1)$: $P(X=1)+P(X=2)=0.4+0.4=0.8$.
3. Expectation: $E(X)=0\times 0.2+1\times 0.4+2\times 0.4=1.2$.

Visual: PMF Graph

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5. Expectation and Variance

• Expectation ($E(X)$): Average value of $X$.
• Variance ($\text{Var}(X)$): Measures spread, $E[(X-\mu {)}^2]=E(X^2)-[E(X){]}^2$.

Example 5.1: Daraz Order Delays

Daraz’s delivery delays ($X$) have PMF: $$P(X)=\{\begin{array}{ll}0.3& X=0\\ 0.5& X=1\\ 0.2& X=2\end{array}$$ Find mean delay and variance.

Solution:

1. Mean: $E(X)=0\times 0.3+1\times 0.5+2\times 0.2=0.9$ days.
2. Variance: $E(X^2)=0+1\times 0.5+4\times 0.2=1.3$. $\text{Var}(X)=1.3-(0.9{)}^2=0.49$.

Visual: Variance Interpretation

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6. Binomial and Normal Distributions

Binomial Distribution

• Scenario: Fixed trials ($n$), success probability ($p$).
• PMF: $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$.
• Example: 70% of chips pass coating. Probability at least 12 pass in 15 trials. P(X \geq 12) = 1 - P(X \leq 11) = 1 - \sum_{k=0}^{11} \binom{15}{k} (0.7)^k (0.3)^{15-k} \approx 0.7158 \text{ (71.58%)}

Normal Distribution

• Properties: Symmetric, bell-shaped, defined by mean ($\mu$) and standard deviation ($\sigma$).
• 68-95-99.7 Rule:
  • 68% within $\mu \pm \sigma$
  • 95% within $\mu \pm 2\sigma$
  • 99.7% within $\mu \pm 3\sigma$

Example 6.1: NTC Call Wait Times

Wait times are $N(\mu =5,\sigma =1.5)$ minutes. Probability a call waits >7 minutes: Z = \frac{7 - 5}{1.5} = 1.33 \Rightarrow P(Z > 1.33) = 1 - \Phi(1.33) \approx 0.0918 \text{ (9.18%)}

Visual: Normal Distribution

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In the Real World

1. eSewa Fraud Detection

   • Idea: Bayes’ Theorem updates fraud probability given an alert.
   • How: If $P(\text{Fraud}|\text{Alert})=33\%$ (from earlier), eSewa uses this to flag high-risk transactions dynamically.

2. Ncell Network Reliability

   • Idea: Binomial Distribution models call failures.
   • How: If 55% of calls fail (from Example 2.1), Ncell uses this to predict outages and reroute traffic.

3. Daraz Delivery Optimization

   • Idea: Expectation and Variance of delays.
   • How: Mean delay = 0.9 days (Example 5.1) helps set customer expectations, while variance (${\sigma }^2=0.49$) guides warehouse stocking.

4. NEPSE Stock Returns

   • Idea: Normal Distribution models returns.
   • How: If daily returns are $N(\mu =0.2\%,\sigma =1.5\%)$, traders use the 68-95-99.7 rule to assess risk (e.g., 95% of returns fall within $-2.8\%$ to $+3.2\%$).

5. Khalti Transaction Limits

   • Idea: Conditional Probability for fraud.
   • How: If $P(\text{Large Transaction}|\text{New User})=5\%$, Khalti may impose stricter limits until user history builds.

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Exam Tip

1. Always check for independence before multiplying probabilities. If events are independent, $P(A\cap B)=P(A)P(B)$.
2. For PMFs, ensure $\sum P(X)=1$ to find unknown constants ($K$).
3. Bayes’ Theorem is tested often—draw a tree diagram to visualize $P(A|B)$.
4. Normal distribution problems require converting to $Z$-scores: $$Z=\frac{X-\mu }{\sigma }$$ Use standard normal tables or calculators.
5. Binomial problems often ask for "at least" or "at most"—use complements to simplify: $$P(X\ge k)=1-P(X\le k-1)$$
6. Real-world ties: Relate problems to eSewa/Khalti (fraud), Ncell/Daraz (delays), or NEPSE (returns) to score application marks.

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Past Exam Patterns:

• 50%: Derive $K$ or compute probabilities from PMFs.
• 30%: Apply Bayes’ Theorem or conditional probability.
• 20%: Binomial/normal distribution problems (e.g., "at least 12 out of 15").