Binomial & Normal Distributions: Models, Calculations & Applications

TAKEAWAYS:

• Binomial distribution models exactly n independent trials with two outcomes (success/failure) and fixed probability p—use it for pass/fail tests, defective items, or customer complaints.
• Normal distribution describes continuous data like heights, exam scores, or battery life—its symmetry and 68-95-99.7 rule simplify probability calculations via Z-scores.
• Standard normal distribution (Z ~ N(0,1)) converts any normal variable to a universal scale for lookup in Z-tables or software.
• Fitting distributions: Compare observed data to theoretical models using mean/variance or goodness-of-fit tests (e.g., χ²).
• Applications: Binomial predicts defective products (e.g., Daraz returns), normal estimates customer wait times (e.g., Ncell call centers) or financial risks (e.g., NEPSE stock returns).
• Exam focus: Calculate probabilities, find parameters (mean/σ), and interpret real-world scenarios using tables or formulas.

1. Binomial Distribution: The Rule of "Yes/No" Trials

Definition and Key Features

The binomial distribution models the number of successes in n independent trials, where:

• Each trial has two outcomes: success (probability p) or failure (probability 1-p).
• Probability p remains constant across trials.
• Trials are independent (one trial’s outcome doesn’t affect another).

Probability Mass Function (PMF): $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k},k=0,1,2,\dots ,n$$ where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$ is the combination formula.

Parameters and Mean/Variance

• Mean (Expected Value): $\mu =np$
• Variance: ${\sigma }^2=np(1-p)$
• Standard Deviation: $\sigma =\sqrt{np(1-p)}$

Worked Example 1: Fitting Binomial Distribution

Problem: A factory produces light bulbs with a 5% defect rate. In a sample of 20 bulbs, fit a binomial distribution to the following observed defects:

Defects (x): 0 | 1 | 2 | 3 | 4
Frequency (f): 28 | 62 | 46 | 10 | 4

Solution:

1. Calculate observed probabilities: Total observations = 28 + 62 + 46 + 10 + 4 = 150. $P(X=0)=28/150\approx 0.187$, $P(X=1)=62/150\approx 0.413$, etc.

2. Assume binomial with $n=20$, $p=0.05$: Compute theoretical probabilities using the PMF: $$P(X=0)=\left(\genfrac{}{}{0ex}{}{20}{0}\right)(0.05{)}^0(0.95{)}^{20}\approx 0.358$$ $$P(X=1)=\left(\genfrac{}{}{0ex}{}{20}{1}\right)(0.05{)}^1(0.95{)}^{19}\approx 0.377$$ (Continue for $k=2,3,4$.)

3. Compare observed vs. theoretical:

   | Defects (x) | Observed (f) | Theoretical P(X=x) | Expected (f) |
   |-------------|---------------|----------------------|--------------|
   | 0           | 28            | 0.358                | 53.7         |
   | 1           | 62            | 0.377                | 56.6         |
   | 2           | 46            | 0.190                | 28.5         |
   | 3           | 10            | 0.058                | 8.7          |
   | 4           | 4             | 0.013                | 2.0          |
   

   Conclusion: The binomial model with $n=20$, $p=0.05$ approximates the data reasonably well (discrepancies may be due to sampling variability).

When to Use Binomial Distribution

flowchart TD
    A["Binomial Distribution"] --> B["Fixed number of trials (n)"]
    A --> C["Two possible outcomes (success/failure)"]
    A --> D["Constant probability of success (p)"]
    A --> E["Independent trials"]
    A --> F["Discrete data (counts)"]
    A --> G["Examples"]
    G --> G1["Defective items in a batch"]
    G --> G2["Pass/fail exams"]
    G --> G3["Customer complaints in a month"]

Real-World Example: Daraz Order Fulfillment

• Scenario: Daraz promises 95% of orders are delivered within 24 hours. For 50 orders, what’s the probability that exactly 45 are delivered on time?
• Solution: $n=50$, $p=0.95$, $k=45$. $$P(X=45)=\left(\genfrac{}{}{0ex}{}{50}{45}\right)(0.95{)}^{45}(0.05{)}^5\approx 0.174$$ Interpretation: There’s a 17.4% chance that exactly 45 out of 50 orders meet Daraz’s delivery promise.

2. Normal Distribution: The Bell Curve of Continuous Data

Definition and Properties

The normal distribution is a continuous probability distribution characterized by:

• Bell-shaped curve symmetric about the mean ($\mu$).
• Asymptotic tails: The curve approaches but never touches the x-axis.
• Parameters:
  • Mean ($\mu$): Center of the distribution.
  • Standard Deviation ($\sigma$): Measures spread (width of the curve).

Probability Density Function (PDF): $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$$

Empirical Rule (68-95-99.7)

• 68% of data falls within $\mu \pm \sigma$.
• 95% within $\mu \pm 2\sigma$.
• 99.7% within $\mu \pm 3\sigma$.

Worked Example 2: Exam Scores (Nepal Board Exam)

Problem: In a TU exam, scores are normally distributed with $\mu =60$ and $\sigma =10$. What percentage of students scored:

1. Between 50 and 70?
2. Below 40?
3. Above 85?

Solution: Convert scores to Z-scores: $$Z=\frac{X-\mu }{\sigma }$$

1. Between 50 and 70:

   • $Z_{50}=(50-60)/10=-1$
   • $Z_{70}=(70-60)/10=+1$
   • From Z-table: $P(-1<Z<1)=0.6826$ or 68.26%.

2. Below 40:

   • $Z_{40}=(40-60)/10=-2$
   • $P(Z<-2)=0.0228$ or 2.28%.

3. Above 85:

   • $Z_{85}=(85-60)/10=+2.5$
   • $P(Z>2.5)=1-P(Z<2.5)=1-0.9938=0.0062$ or 0.62%.

Standard Normal Distribution

• Any normal distribution $X\sim N(\mu ,{\sigma }^2)$ can be converted to the standard normal $Z\sim N(0,1)$ using: $$Z=\frac{X-\mu }{\sigma }$$
• Use Z-tables or software (e.g., Excel’s NORM.S.DIST) to find probabilities.

Worked Example 3: Ncell Battery Life

Problem: Ncell claims their phone batteries last an average of 50 hours ($\mu =50$) with $\sigma =15$ hours. What’s the probability a battery lasts:

1. Less than 30 hours?
2. Between 40 and 60 hours?

Solution:

1. Less than 30 hours: $$Z=\frac{30-50}{15}=-1.33$$ $P(Z<-1.33)\approx 0.0918$ or 9.18%.

2. Between 40 and 60 hours: $$Z_{40}=\frac{40-50}{15}=-0.67,Z_{60}=\frac{60-50}{15}=+0.67$$ $P(-0.67<Z<0.67)=0.4522+0.4522=0.9044$ or 90.44%.

When to Use Normal Distribution

flowchart TD
    A["Normal Distribution"] --> B["Continuous data"]
    A --> C["Bell-shaped and symmetric"]
    A --> D["Known mean (μ) and standard deviation (σ)"]
    A --> E["Examples"]
    E --> E1["Exam scores (TU, PU)"]
    E --> E2["Customer wait times (Pathao, Ncell)"]
    E --> E3["Manufacturing defects (Daraz quality control)"]
    E --> E4["Stock prices (NEPSE)"]

Real-World Example: NEPSE Stock Returns

• Scenario: NEPSE stock returns are normally distributed with $\mu =8\%$ and $\sigma =4\%$. What’s the probability of a return:
  1. Below 0% (loss)?
  2. Between 4% and 12%?
• Solution:
  1. $Z=(0-8)/4=-2$ $P(Z<-2)=0.0228$ or 2.28% chance of a loss.
  2. $Z_{4\%}=(4-8)/4=-1$, $Z_{12\%}=(12-8)/4=+1$ $P(-1<Z<1)=0.6826$ or 68.26% chance of returns in this range.

3. Comparing Binomial and Normal Distributions

Worked Example 4: Approximating Binomial with Normal

Problem: A call center receives 100 calls/day with a 10% chance of being complaints. Approximate the probability of 15 or more complaints using the normal distribution.

Solution:

1. Binomial Parameters: $n=100$, $p=0.1$, $\mu =np=10$, $\sigma =\sqrt{np(1-p)}=3$.

2. Continuity Correction: For $P(X\ge 15)$, use $P(X\ge 14.5)$ in the normal approximation.

3. Z-Score: $$Z=\frac{14.5-10}{3}=1.5$$ $P(Z\ge 1.5)=1-P(Z<1.5)=1-0.9332=0.0668$ or 6.68%.

Note: The exact binomial probability is $P(X\ge 15)=0.0665$, so the approximation is very close.

4. Applications in Nepal’s Context

Example 1: Khalti Transaction Success Rate

• Scenario: Khalti claims 98% of transactions succeed. For 200 transactions, what’s the probability of fewer than 190 successes?
• Solution: Binomial with $n=200$, $p=0.98$, $k=189$ (since $P(X<190)=P(X\le 189)$). $$P(X\le 189)=1-P(X\ge 190)\approx 1-0.0026=0.9974\text{ (using normal approximation)}$$ Interpretation: Extremely unlikely (0.26%) to have fewer than 190 successes in 200 transactions.

Example 2: NTC Internet Speed

• Scenario: NTC advertises average internet speed of 50 Mbps ($\mu =50$) with $\sigma =10$ Mbps. What’s the probability a user gets less than 30 Mbps?
• Solution: $$Z=\frac{30-50}{10}=-2$$ $P(Z<-2)=0.0228$ or 2.28% chance of slow speeds.

Example 3: TU Exam Pass Rates

• Scenario: TU exams have a pass rate of 60% ($\mu =60$, $\sigma =15$ for a class of 100 students). How many students are expected to score above 80?
• Solution: $$Z=\frac{80-60}{15}=1.33$$ $P(Z>1.33)=1-0.9082=0.0918$ or 9.18 students.

5. Key Formulas Summary

Exam Tip

1. Binomial Questions:

   • Always check if the scenario fits binomial assumptions (fixed n, constant p, independent trials).
   • Use the PMF formula or tables for small n; approximate with normal for large n (with continuity correction).
   • Common pitfalls: Forgetting to adjust for continuity when approximating binomial with normal.

2. Normal Distribution Questions:

   • Always standardize to Z-scores for probabilities.
   • Memorize the empirical rule (68-95-99.7) for quick estimates.
   • For "above/below" questions, draw the curve and shade the relevant area.
   • Common pitfalls: Misapplying the empirical rule (e.g., thinking 95% is within $\mu \pm 1.96\sigma$, not $\mu \pm 2\sigma$).

3. Real-World Scenarios:

   • Translate word problems into statistical terms (e.g., "defective items" → binomial, "exam scores" → normal).
   • Use given percentages to infer $\mu$ and $\sigma$ (e.g., if 10% are below 20, assume $Z=-1.28$ for $\mu$ and $\sigma$).

4. Graphs and Tables:

   • Sketch the distribution (binomial as bars, normal as a bell curve) and shade the required area.
   • For binomial, label the x-axis with possible counts (0 to n).
   • For normal, label $\mu$ and $\mu \pm \sigma$, $\mu \pm 2\sigma$, etc.

5. Calculations:

   • Use calculators or software (e.g., Excel, Python) for complex probabilities.
   • For Z-tables, remember:
     • $P(Z<a)=1-P(Z<-a)$ (symmetry).
     • $P(a<Z<b)=P(Z<b)-P(Z<a)$.

Final Note: Practice fitting distributions to data and interpreting probabilities in context. The key to full marks is clear steps, correct formulas, and real-world relevance. Always ask: "Does this scenario fit binomial or normal? What’s the probability we’re solving for?"