Probability theory is essential for cooperative managers in Kenya who must make decisions under uncertainty, such as assessing loan default risks or forecasting member participation in meetings. Understanding probability events and their types enables managers to quantify risks and opportunities effectively. This chapter focuses on applying basic probability theory through the study of different event types, equipping cooperative professionals with skills to analyze uncertain outcomes confidently.

7.1 Probability events

Probability events describe possible outcomes of an uncertain situation, fundamental to risk assessment in cooperatives. An event is any subset of all possible outcomes (the sample space) of a random experiment. The probability of an event is a measure between 0 and 1 indicating the likelihood of that event occurring.

7.1.1 Definition of an event

An event is any collection of outcomes from a random experiment. The sample space \( S \) is the set of all possible outcomes.

The probability of an event \( A \), denoted \( P(A) \), satisfies:

$$ 0 \leq P(A) \leq 1 $$

$$ P(S) = 1 $$

7.1.2 Sample space and outcomes

The sample space \( S \) contains all possible outcomes of an experiment.

7.1.3 Probability of an event formula

The probability of event \( A \) is:

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

This applies when all outcomes are equally likely.

7.1.4 Complement of an event

The complement of event \( A \), denoted \( A^c \), is the event that \( A \) does not occur.

The probability of the complement is:

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

Worked Examples

Example 1: A cooperative has 50 members, 20 of whom are women. What is the probability that a randomly selected member is a woman?

Given: Number of women = 20, Total members = 50

$$ P(\text{woman}) = \frac{20}{50} $$

$$ = 0.4 $$

Answer: 0.4

Example 2: In a savings cooperative, the probability that a member defaults on a loan is 0.1. What is the probability that a member does not default?

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

$$ P(\text{no default}) = 1 - 0.1 $$

$$ = 0.9 $$

Answer: 0.9

Example 3: A cooperative holds a raffle with 200 tickets. If one ticket is drawn at random, what is the probability of winning if a member holds 5 tickets?

Given: Favorable outcomes = 5, Total outcomes = 200

$$ P(\text{win}) = \frac{5}{200} $$

$$ = 0.025 $$

Answer: 0.025

Example 4: In a cooperative election, there are 4 candidates. What is the probability that a randomly chosen vote goes to Candidate A?

Given: Number of candidates = 4, assuming equal likelihood

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

$$ = 0.25 $$

Answer: 0.25

Example 5: The probability that a cooperative member attends a meeting is 0.7. Find the probability that the member does not attend.

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

$$ P(\text{not attend}) = 1 - 0.7 $$

$$ = 0.3 $$

Answer: 0.3

Chapter Summary

This chapter introduced the concept of probability events as the foundation for understanding uncertainty in business contexts. It explained different types of events including simple, compound, mutually exclusive, independent, and dependent events, highlighting their distinct characteristics. The fundamental rules of probability were presented to enable calculation of event likelihoods accurately. Bayes' Theorem was covered to demonstrate how prior knowledge can be updated with new information for conditional probability assessments. The chapter also showed how to use probability trees as visual tools to organize and compute complex probability scenarios. Finally, practical applications of probability were explored to illustrate its relevance in decision-making processes within various business environments.