Probability Distributions: Types, Formulas & Business Uses

TAKEAWAYS:

• Probability distributions show how likely different outcomes are in business decisions (e.g., sales, defects, or profits).
• Binomial is for fixed trials with two outcomes (success/failure), Poisson for rare events over time, and Normal for symmetric data like heights or test scores.
• Formulas like $P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$ (binomial) or $\mu =np$ (mean) help calculate probabilities.
• Business uses include risk assessment, quality control, and forecasting (e.g., predicting defective products or customer arrivals).
• Graphs (bar charts for discrete, curves for continuous) visualize distributions—key for interpreting results.
• NEB exams test definitions, formula applications, and real-world scenario problems (e.g., "A factory has 5% defective items; find the probability of 2 defects in 20 items").

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What is a Probability Distribution?

A probability distribution is a function that gives the probabilities of all possible outcomes of a random experiment. It answers:

• What outcomes can happen?
• How likely is each outcome?

Types of Distributions

Probability distributions are classified into two main types:

1. Discrete Distributions

   • Outcomes are countable (e.g., number of customers, defective items).
   • Examples: Binomial, Poisson.
   • Graph: Bar chart (height = probability).

2. Continuous Distributions

   • Outcomes are measurable (e.g., height, weight, time).
   • Examples: Normal, Exponential.
   • Graph: Smooth curve (area under curve = probability).

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1. Binomial Distribution

Definition

• Used when:
  • There are fixed trials (e.g., 10 coin tosses).
  • Each trial has two outcomes (success/failure, e.g., pass/fail).
  • Probability of success ($p$) is constant for each trial.
  • Trials are independent (one trial doesn’t affect another).

Formula

The probability of getting exactly $k$ successes in $n$ trials is: $$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$$ Where:

• $\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$ (combinations).
• $p$ = probability of success.
• $1-p$ = probability of failure.

Key Parameters

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

Worked Example

Problem: A factory produces light bulbs with a 5% defect rate. If 20 bulbs are tested, what is the probability that exactly 2 are defective? Solution:

1. Identify parameters:
   • $n=20$ (trials),
   • $k=2$ (defective bulbs),
   • $p=0.05$ (probability of defect).
2. Use the binomial formula: $$P(X=2)=\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.05{)}^2(0.95{)}^{18}$$
3. Calculate combinations: $$\left(\genfrac{}{}{0ex}{}{20}{2}\right)=\frac{20!}{2!18!}=190$$
4. Calculate probabilities: $$(0.05{)}^2=0.0025$$ $$(0.95{)}^{18}\approx 0.3386\text{(use calculator)}$$
5. Multiply: $$P(X=2)=190\times 0.0025\times 0.3386\approx 0.1608\text{ or }16.08\%$$

Answer: The probability is 16.08%.

Graph of Binomial Distribution

Applications in Business

• Quality control (e.g., probability of defective products).
• Marketing (e.g., probability of customers buying a product).
• Finance (e.g., probability of loan defaults).

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2. Poisson Distribution

Definition

• Used for rare events occurring in a fixed interval (time, distance, area).
• Example: Number of calls to a call center per hour, machine failures per day.
• Assumes events are independent and occur at a constant average rate ($\lambda$).

Formula

The probability of exactly $k$ events occurring is: $$P(X=k)=\frac{e^{-\lambda }{\lambda }^k}{k!}$$ Where:

• $e$ = Euler’s number (~2.71828),
• $\lambda$ = average number of events per interval.

Key Parameters

• Mean ($\mu$): $\lambda$
• Variance (${\sigma }^2$): $\lambda$

Worked Example

Problem: A bank receives an average of 3 complaints per day. What is the probability of receiving exactly 5 complaints tomorrow? Solution:

1. Identify $\lambda =3$ (average complaints/day).
2. Use the Poisson formula for $k=5$: $$P(X=5)=\frac{e^{-3}{3}^5}{5!}$$
3. Calculate: $$e^{-3}\approx 0.0498$$ $$3^5=243$$ $$5!=120$$
4. Multiply: $$P(X=5)=\frac{0.0498\times 243}{120}\approx 0.1008\text{ or }10.08\%$$

Answer: The probability is 10.08%.

Graph of Poisson Distribution

Comparison: Binomial vs. Poisson

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3. Normal Distribution

Definition

• Continuous distribution shaped like a bell curve.
• Symmetric about the mean ($\mu$).
• Used for data like heights, test scores, or measurement errors.
• Defined by mean ($\mu$) and standard deviation ($\sigma$).

Formula

The probability density function (PDF) is: $$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}$$ For probabilities, we use the standard normal table (Z-table): $$Z=\frac{X-\mu }{\sigma }$$

Worked Example

Problem: The heights of 12th-grade students are normally distributed with $\mu =160$ cm and $\sigma =10$ cm. What is the probability that a randomly selected student is taller than 175 cm? Solution:

1. Calculate $Z$-score: $$Z=\frac{175-160}{10}=1.5$$
2. Look up $P(Z>1.5)$ in the Z-table:
   • $P(Z\le 1.5)=0.9332$
   • $P(Z>1.5)=1-0.9332=0.0668$ or 6.68%.

Answer: The probability is 6.68%.

Graph of Normal Distribution

Applications in Business

• Quality control (e.g., tolerances in manufacturing).
• Finance (e.g., stock price movements).
• Human resources (e.g., employee performance scores).

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4. Other Distributions (Brief Overview)

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Exam Tip: How to Score Full Marks

1. Understand Definitions:

   • NEB often asks: "Define binomial distribution" or "When is Poisson used?"
   • Example Answer:

     "Binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with two possible outcomes (success/failure)."

2. Show All Steps in Calculations:

   • Write down the formula, substitute values, and simplify.
   • Example: For $P(X=2)$ in binomial, show: $$\left(\genfrac{}{}{0ex}{}{20}{2}\right)(0.05{)}^2(0.95{)}^{18}\to 190\times 0.0025\times 0.3386$$

3. Draw Graphs When Asked:

   • Sketch bar charts for binomial/Poisson or curves for normal.
   • Label axes and key points (e.g., mean, probability).

4. Apply to Real-World Scenarios:

   • NEB loves word problems. Translate words into parameters ($n$, $p$, $\lambda$).
   • Example Question:

     "A company’s past data shows 10% of orders are delayed. If 15 orders are placed, find the probability of exactly 3 delays." Solution: Binomial with $n=15$, $k=3$, $p=0.1$.

5. Memorize Key Formulas:

   • Binomial: $\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$
   • Poisson: $\frac{e^{-\lambda }{\lambda }^k}{k!}$
   • Normal: $Z=\frac{X-\mu }{\sigma }$

6. Practice Z-Table Lookups:

   • For normal distribution, always convert to $Z$ and use the table.

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NEB Board-Style Questions

Section A: Short Answer (1 mark each)

1. Define probability distribution.
2. State the conditions for a binomial distribution.
3. What is the mean of a Poisson distribution?
4. Write the formula for the standard normal variable $Z$.
5. Give one real-life example of a normal distribution.

Section B: Long Answer (5–7 marks)

1. A factory produces packets of biscuits. The probability of a packet being underweight is 0.05. If 20 packets are selected at random, find the probability that:

   • Exactly 2 packets are underweight.
   • More than 3 packets are underweight. (Use binomial distribution.)

2. The number of accidents in a factory follows a Poisson distribution with an average of 2 accidents per month. Find the probability that:

   • There will be no accidents next month.
   • There will be at least 3 accidents next month.

3. The heights of students in a school are normally distributed with a mean of 150 cm and a standard deviation of 10 cm. Find the probability that a randomly selected student has a height:

   • Between 140 cm and 160 cm.
   • More than 170 cm.

4. Explain the difference between binomial and Poisson distributions with examples. When would you use each?

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Summary Table: Which Distribution to Use?

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Final Notes

• Binomial is for "yes/no" scenarios with fixed trials.
• Poisson is for counting rare events (e.g., calls, defects).
• Normal is for continuous data like heights or test scores.
• Always check if the problem involves discrete (binomial/Poisson) or continuous (normal) data.
• Practice converting word problems into mathematical terms (e.g., "probability of success" = $p$).

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Bell curve labeled with μ and σ. (Image: Inductiveload, Public domain, via Wikimedia Commons)