By the end of this chapter, you will be able to:
These skills will help you make smart, data-driven decisions in real business situations, boosting your confidence and effectiveness in the trade.
Compiled by: Susan Mburu
Probability theory is essential in cooperative management for decision-making under uncertainty. Managers often face situations such as predicting loan defaults, estimating member participation, or assessing risks in cooperative investments. This chapter introduces the basic concepts of probability events and their types, enabling cooperative professionals to apply quantitative reasoning in evaluating uncertain outcomes effectively.
Probability events describe outcomes or sets of outcomes from random experiments relevant to cooperatives, such as member loan approvals or product demand fluctuations. Understanding probability events helps quantify the likelihood of various scenarios, improving risk assessment and strategic planning.
A probability event is any outcome or set of outcomes from a random experiment. The probability of an event \(E\) is a number between 0 and 1 that measures how likely it is to occur.
$$ 0 \leq P(E) \leq 1 $$
Example 1: A cooperative has 100 members, 40 of whom have active loans. What is the probability that a member selected at random has an active loan?
Given: Total members \(N=100\), members with active loans \(n=40\)$$P(E) = \frac{n}{N}$$$$P(E) = \frac{40}{100}$$$$P(E) = 0.4$$
Answer: 0.4 (40%)
Example 2: A cooperative store sells 3 types of products: maize, beans, and rice. Maize is bought 50 times, beans 30 times, rice 20 times in a day. What is the probability a randomly chosen sale is maize?
Given: Total sales \(N=50+30+20=100\), maize sales \(n=50\)$$P(E) = \frac{50}{100} = 0.5$$
Answer: 0.5 (50%)
Example 3: In a cooperative election, 5 candidates compete. A member votes for one candidate at random. What is the probability the vote goes to candidate A?
Given: Number of candidates \(N=5\), candidate A = 1 choice$$P(E) = \frac{1}{5} = 0.2$$
Answer: 0.2 (20%)
The sample space \(S\) is the set of all possible outcomes of an experiment. An event is a subset of the sample space.
$$ S = \{s_1, s_2, ..., s_n\} $$
Example 1: A cooperative randomly selects one member from 3 branches: Nairobi, Kisumu, Eldoret. What is the sample space?
Given: Branches = Nairobi, Kisumu, Eldoret$$S = \{\text{Nairobi}, \text{Kisumu}, \text{Eldoret}\}$$
Answer: Sample space with 3 outcomes
Example 2: In a loan approval process, outcomes are Approve (A) or Reject (R). What is the sample space?$$S = \{A, R\}$$ Answer: Sample space with 2 outcomes
Example 3: A cooperative's product quality test can result in Pass (P), Fail (F), or Rework (R). Sample space is:$$S = \{P, F, R\}$$ Answer: Sample space with 3 outcomes
The probability of event \(E\) is the ratio of favorable outcomes to total outcomes in sample space \(S\).
$$ P(E) = \frac{|E|}{|S|} $$
Example 1: In a cooperative with 10 products, 3 are defective. What is the probability a randomly chosen product is defective?
Given: Defective \(=3\), total \(=10\)$$P(E) = \frac{3}{10} = 0.3$$
Answer: 0.3 (30%)
Example 2: A cooperative committee of 4 members is chosen from 10 candidates. Probability that a specific member is selected in a random pick?
Given: Favorable outcomes \(=1\), total \(=10\)$$P(E) = \frac{1}{10} = 0.1$$
Answer: 0.1 (10%)
Example 3: Out of 50 cooperative loans, 5 defaulted. Probability a randomly chosen loan defaults:$$P(E) = \frac{5}{50} = 0.1$$ Answer: 0.1 (10%)
The complement of event \(E\), denoted \(E^c\), is the event that \(E\) does not occur.
$$ P(E^c) = 1 - P(E) $$
Example 1: Probability a member does not have an active loan if \(P(\text{active loan})=0.4\)$$P(E^c) = 1 - 0.4 = 0.6$$ Answer: 0.6 (60%)
Example 2: Probability a product passes quality test if failure probability is 0.2$$P(E^c) = 1 - 0.2 = 0.8$$ Answer: 0.8 (80%)
Example 3: Probability a loan is not defaulted if default probability is 0.1$$P(E^c) = 1 - 0.1 = 0.9$$ Answer: 0.9 (90%)
Create a free account to open more of this chapter.
Free: practical guides, quick cards, workplace scenarios and more.
Create a free accountThis chapter introduced the concept of probability events as the foundation for understanding uncertainty in business contexts. It examined different types of events including simple events, compound events, mutually exclusive events, independent events, and dependent events, highlighting their distinct characteristics. The chapter then explored the fundamental rules of probability that govern how probabilities are combined and calculated. Bayes' Theorem was presented as a powerful tool for updating probabilities based on new information. Methods for drawing probability trees were explained to visually represent sequential events and their probabilities. Finally, the chapter demonstrated practical applications of probability theory in decision-making and risk assessment within various business scenarios. This comprehensive approach equips learners to analyze and interpret uncertain outcomes systematically.
A cooperative has 50 members. If one member is selected at random, what is the probability that the member is a woman, given that 30 members are women? (2 marks)
In a savings cooperative, the probability that a member defaults on loan repayment is 0.1. What is the probability that a randomly selected member does not default? (2 marks)
At the start of this chapter we promised you would be able to:
Tick each one you can genuinely do.
So, are you there yet?
You're competent when you can confidently do 50% or more of what this chapter promised.
Sign in to record how you're doing.