Probability theory is fundamental in decision-making across all professional fields in Kenya. Whether in healthcare, agriculture, business, or government, understanding the likelihood of events helps professionals manage risks, forecast outcomes, and optimize resources. This chapter introduces basic probability concepts, enabling diploma students to apply probability calculations effectively in diverse real-world scenarios. Mastery of these principles equips students to analyze uncertainty and make informed judgments.

7.1 Probability events

Probability events form the foundation of probability theory. An event is any outcome or set of outcomes of a random experiment, such as the chance of rainfall affecting a farm’s harvest or the likelihood of a client defaulting on a loan at a SACCO. Calculating the probability of events enables professionals to quantify uncertainty and plan accordingly.

7.1.1 Definition of an event

An event is a specific outcome or a set of outcomes from a random experiment. For example, in a hospital, the event could be "a patient recovering within 7 days." The probability of an event is a number between 0 and 1, where 0 means the event cannot occur and 1 means it is certain.

The probability \(P(E)\) of an event \(E\) is given by

$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$

Worked Examples

Example 1: A university library has 5 new books and 15 old books. What is the probability that a randomly selected book is new?

Given: Number of new books = 5, Total books = 5 + 15 = 20

$$ P(\text{new book}) = \frac{5}{20} $$

$$ = 0.25 $$

Answer: 0.25 (25%)

Example 2: A retail shop has 8 red shirts and 12 blue shirts. What is the probability that a randomly selected shirt is blue?

Given: Number of blue shirts = 12, Total shirts = 8 + 12 = 20

$$ P(\text{blue shirt}) = \frac{12}{20} $$

$$ = 0.6 $$

Answer: 0.6 (60%)

Example 3: A farmer has 10 cows, 7 goats, and 3 sheep. What is the probability that an animal selected at random is a goat?

Given: Number of goats = 7, Total animals = 10 + 7 + 3 = 20

$$ P(\text{goat}) = \frac{7}{20} $$

$$ = 0.35 $$

Answer: 0.35 (35%)

7.1.2 Sample space and event sets

The sample space \(S\) is the set of all possible outcomes in an experiment. An event \(E\) is a subset of \(S\). For example, in a county government office, if an employee is selected at random from departments A, B, and C, the sample space is \(\{A, B, C\}\).

The number of possible outcomes in \(S\) is denoted \(n(S)\), and the number of favorable outcomes in \(E\) is \(n(E)\).

$$ P(E) = \frac{n(E)}{n(S)} $$

Worked Examples

Example 1: A hotel has 4 types of rooms: single, double, suite, and presidential. What is the probability that a randomly booked room is a suite?

Given: \(S = \{single, double, suite, presidential\}\), \(n(S) = 4\), \(n(E) = 1\)

$$ P(\text{suite}) = \frac{1}{4} $$

$$ = 0.25 $$

Answer: 0.25 (25%)

Example 2: A SACCO has members from 3 age groups: 18-30, 31-45, and over 45. If a member is selected at random, what is the probability they are aged 31-45?

Given: \(S = \{18-30, 31-45, >45\}\), \(n(S) = 3\), \(n(E) = 1\)

$$ P(31-45) = \frac{1}{3} $$

$$ = 0.333 $$

Answer: 0.333 (33.3%)

Example 3: A farmer plants 5 types of crops. What is the probability that a crop randomly selected for inspection is maize?

Given: \(S = 5\) crop types, maize is 1 type

$$ P(\text{maize}) = \frac{1}{5} $$

$$ = 0.2 $$

Answer: 0.2 (20%)

7.1.3 Probability scale and interpretation

Probability values lie between 0 and 1. A probability of 0 means the event cannot happen, while a probability of 1 means it will certainly happen. Values between indicate the likelihood of occurrence.

• \(P(E) = 0\): Impossible event
• \(0 < P(E) < 0.5\): Unlikely event
• \(P(E) = 0.5\): Even chance
• \(0.5 < P(E) < 1\): Likely event
• \(P(E) = 1\): Certain event

Worked Examples

Example 1: A county referral hospital has a 0.1 probability of equipment failure in a day. Interpret this probability.

Given: \(P(\text{failure}) = 0.1\)

• Since 0.1 < 0.5, equipment failure is unlikely on any given day.

Example 2: A retailer knows that 0.7 of customers buy a product after a demonstration. Interpret this.

Given: \(P(\text{purchase}) = 0.7\)

• Since 0.7 > 0.5, purchase is likely after demonstration.

Example 3: A cooperative forecasts a 0.5 chance of rainfall tomorrow. Interpret.

Given: \(P(\text{rain}) = 0.5\)

• Rain and no rain are equally likely.

7.1.4 Complement of an event

The complement of an event \(E\), written \(E^c\), is the event that \(E\) does not occur. The sum of the probabilities of an event and its complement is always 1.

$$ P(E^c) = 1 - P(E) $$

Worked Examples

Example 1: A bank finds that the probability a customer defaults on a loan is 0.2. What is the probability a customer does not default?

Given: \(P(\text{default}) = 0.2\)

$$ P(\text{no default}) = 1 - 0.2 $$

$$ = 0.8 $$

Answer: 0.8 (80%)

Example 2: A hotel notes that the probability of a room being occupied is 0.65. Find the probability that the room is vacant.

Given: \(P(\text{occupied}) = 0.65\)

$$ P(\text{vacant}) = 1 - 0.65 $$

$$ = 0.35 $$

Answer: 0.35 (35%)

Example 3: A NEMA report states the probability of a hazardous waste spill is 0.03. Find the probability of no spill.

Given: \(P(\text{spill}) = 0.03\)

$$ P(\text{no spill}) = 1 - 0.03 $$

$$ = 0.97 $$

Answer: 0.97 (97%)

Chapter Summary

This chapter introduced the concept of probability events as the foundation for understanding chance in various contexts. It explained different types of events including simple events with a single outcome, compound events combining multiple outcomes, mutually exclusive events that cannot occur together, independent events where the occurrence of one does not affect the other, and dependent events where one event influences the probability of another. The chapter then covered the fundamental rules of probability that govern how probabilities are calculated and combined. Bayes' Theorem was presented as a method to update probabilities based on new information. The use of probability trees was demonstrated to visually organize and calculate probabilities of sequential events. Finally, the chapter explored practical applications of probability theory in solving real-world problems, emphasizing its importance in decision making and risk assessment.