Probability: Basic Concepts, Addition and Multiplication Theorems, and Bayes' Theorem

Introduction to Probability

Probability is the branch of mathematics that deals with the numerical description of how likely an event is to occur. In the hospitality industry, probability is used for forecasting room occupancy, managing food waste, and assessing the risk of flight delays for arriving guests.

Basic Terminology

Before calculating probabilities, we must understand the building blocks:

1. Random Experiment: An action or process that leads to one of several possible outcomes, where the exact outcome cannot be predicted with certainty.
   • Example: Tossing a coin or checking if a guest will check out on time.
2. Sample Space ($S$): The set of all possible outcomes of a random experiment.
   • Example: For a single die roll, $S=\{1,2,3,4,5,6\}$.
3. Event ($E$): A subset of the sample space. It is a specific outcome or a collection of outcomes we are interested in.
   • Example: Rolling an even number, $E=\{2,4,6\}$.

A standard 52-card deck used to demonstrate sample spaces and events. (Image: AustinGabriel64, CC0, via Wikimedia Commons)

Approaches to Probability

There are three primary ways to assign probability to an event:

Worked Example 1: Classical Probability

Question: A hotel manager has a bowl containing 5 red, 3 blue, and 2 green key cards. If one card is picked at random, what is the probability that it is either blue or green?

Step-by-step Solution:

1. Identify the Sample Space ($S$): Total cards = $5+3+2=10$. So, $n(S)=10$.
2. Identify the Event ($E$): The event is picking a blue or green card.
   • Number of blue cards = 3
   • Number of green cards = 2
   • Total favorable outcomes $n(E)=3+2=5$.
3. Apply the Formula: $$P(E)=\frac{n(E)}{n(S)}=\frac{5}{10}=0.5\text{ or }50\%$$

Types of Events

Understanding the relationship between events is crucial for choosing the right theorem.

Addition Theorem of Probability

The Addition Theorem is used when we want to find the probability of event $A$ OR event $B$ occurring ($A\cup B$).

1. Mutually Exclusive Events

Events are mutually exclusive if they cannot occur at the same time. Formula: $$P(A\cup B)=P(A)+P(B)$$

2. Non-Mutually Exclusive Events

Events are non-mutually exclusive if they can occur simultaneously. To avoid double-counting the overlap, we subtract the intersection. Formula: $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Worked Example 2: Non-Mutually Exclusive Events

Question: In a hotel staff of 100 employees, 60 speak English, 40 speak Nepali, and 20 speak both. If an employee is chosen at random, what is the probability they speak English or Nepali?

Step-by-step Solution:

1. Define Events:
   • $P(A)$: Probability of speaking English = $\frac{60}{100}=0.6$
   • $P(B)$: Probability of speaking Nepali = $\frac{40}{100}=0.4$
   • $P(A\cap B)$: Probability of speaking both = $\frac{20}{100}=0.2$
2. Apply Addition Theorem: $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$ $$P(A\cup B)=0.6+0.4-0.2=0.8$$
3. Result: The probability is 0.8 or 80%.

Conditional Probability and Multiplication Theorem

Conditional Probability

Conditional probability is the probability of an event $A$ occurring, given that event $B$ has already occurred. It is denoted by $P(A|B)$.

Formula: $$P(A|B)=\frac{P(A\cap B)}{P(B)},\text{ where }P(B)>0$$

Multiplication Theorem

This theorem is used to find the probability of event $A$ AND event $B$ occurring ($A\cap B$).

1. For Independent Events: The outcome of one does not affect the other. $$P(A\cap B)=P(A)\times P(B)$$
2. For Dependent Events: The outcome of the first affects the second. $$P(A\cap B)=P(A)\times P(B|A)$$

Worked Example 3: Dependent Events

Question: A restaurant has 10 dinner plates, 3 of which are chipped. If a waiter picks 2 plates one after another without replacement, what is the probability that both are chipped?

Step-by-step Solution:

1. First Event ($A$): Picking the first chipped plate. $$P(A)=\frac{3}{10}$$
2. Second Event ($B|A$): Picking the second chipped plate given the first was chipped.
   • Remaining chipped plates = 2
   • Remaining total plates = 9 $$P(B|A)=\frac{2}{9}$$
3. Apply Multiplication Theorem: $$P(A\cap B)=P(A)\times P(B|A)=\frac{3}{10}\times \frac{2}{9}=\frac{6}{90}=\frac{1}{15}\approx 0.0667$$

Bayes' Theorem

Bayes' Theorem is a powerful tool used to update the probability of a hypothesis as more evidence or information becomes available. It is essentially the "reverse" of conditional probability.

Formula: $$P(A|B)=\frac{P(B|A)\cdot P(A)}{P(B)}$$ Where $P(B)$ (the total probability) is often calculated as: $$P(B)=\sum [P(B|A_i)\cdot P(A_i)]$$

Worked Example 4: Bayes' Theorem in Hotel Management

Question: A hotel receives bookings from three sources: Online (60%), Travel Agent (30%), and Direct (10%). The probability of a "No-Show" (guest doesn't arrive) is 5% for Online, 2% for Agents, and 1% for Direct bookings. If a guest is a "No-Show", what is the probability they booked Online?

Step-by-step Solution:

1. Prior Probabilities:
   • $P(\text{Online})=0.60$
   • $P(\text{Agent})=0.30$
   • $P(\text{Direct})=0.10$
2. Conditional Probabilities (Likelihood of No-Show):
   • $P(\text{No-Show}|\text{Online})=0.05$
   • $P(\text{No-Show}|\text{Agent})=0.02$
   • $P(\text{No-Show}|\text{Direct})=0.01$
3. Total Probability of No-Show $P(B)$: $$P(\text{No-Show})=(0.60\times 0.05)+(0.30\times 0.02)+(0.10\times 0.01)$$ $$P(\text{No-Show})=0.030+0.006+0.001=0.037$$
4. Apply Bayes' Theorem: $$P(\text{Online}|\text{No-Show})=\frac{P(\text{No-Show}|\text{Online})\times P(\text{Online})}{P(\text{No-Show})}$$ $$P(\text{Online}|\text{No-Show})=\frac{0.05\times 0.60}{0.037}=\frac{0.030}{0.037}\approx 0.8108$$ Result: There is an 81.08% probability that the no-show guest booked online.

In the real world

Probability is not just a classroom exercise; it is the engine behind many services in Nepal:

1. Insurance Companies (e.g., Nepal Insurance Company): They use Empirical Probability and actuarial tables to determine premiums. If historical data shows a 2% probability of a hotel fire in a certain area, the premium is calculated based on that risk.
2. Ride-Sharing Apps (Pathao/InDrive): These apps use Conditional Probability to estimate "Time of Arrival" (ETA). The probability of a driver arriving in 5 minutes is conditional upon the current traffic density in Kathmandu (e.g., $P(\text{Arrival in 5min}|\text{Traffic is Heavy})$).
3. E-commerce (Daraz): Daraz uses probability for its recommendation engine. If you bought a hotel towel, the probability that you might need a bath mat is higher. This is a form of Joint Probability used in "Frequently Bought Together" sections.

Exam tip

When solving probability questions in the TU board exam, follow these steps to ensure full marks:

1. Define your events clearly: Always start by writing "Let $A=\dots$ and $B=\dots$".
2. State the formula: Write the general formula (e.g., the Addition Theorem) before plugging in the numbers.
3. Check for Mutual Exclusivity: Before using $P(A)+P(B)$, explicitly state whether the events are mutually exclusive or not.
4. Simplify Fractions: Always provide the final answer in both fraction and decimal/percentage form.
5. Bayes' Theorem Layout: For Bayes' problems, create a small table for Prior and Conditional probabilities first; this prevents calculation errors in the total probability denominator.