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Compiled by: Samuel Maina
Probability theory is essential in decision-making across all professional fields in Kenya. Whether managing risks in a county referral hospital, forecasting sales in a retail business, or assessing quality control in a manufacturing cooperative, understanding probability enables professionals to quantify uncertainty and make informed choices. This chapter introduces fundamental concepts of probability, including events, outcomes, and sample spaces, laying the groundwork for more advanced probability applications in business mathematics and statistics.
Probability measures the likelihood of an event occurring within a defined set of possible outcomes. It quantifies uncertainty, allowing professionals in sectors such as banking, agriculture, and hospitality to evaluate risks and opportunities. This section covers the basic terminology and mathematical formulations used in probability theory.
Probability quantifies how likely an event is to occur and is expressed as a number between 0 and 1, where 0 means impossibility and 1 means certainty. For an event \(E\) occurring within a sample space \(S\), the probability \(P(E)\) is calculated by the ratio of the number of favorable outcomes to the total number of possible outcomes. This is expressed as:
$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$
Example 1: A SACCO issues membership cards with serial numbers from 1 to 100. What is the probability that a randomly selected card number is a multiple of 5?
Given: Total cards \(= 100\), favorable outcomes \(= \) multiples of 5 from 1 to 100.
Number of multiples of 5 between 1 and 100:
$$ \frac{100}{5} = 20 $$
Total outcomes \(= 100\)
$$ P(E) = \frac{20}{100} $$
$$ = 0.2 $$
Answer: 0.2 (20% chance)
Example 2: A county hospital has 10 nurses, 4 of whom are trained in emergency care. If a nurse is randomly selected, what is the probability the nurse is trained in emergency care?
Given: Total nurses \(= 10\), favorable outcomes \(= 4\).
$$ P(E) = \frac{4}{10} $$
$$ = 0.4 $$
Answer: 0.4 (40% chance)
Example 3: A hotel offers 3 types of rooms: standard, deluxe, and suite. There are 15 standard rooms, 10 deluxe, and 5 suites. What is the probability that a randomly selected room is deluxe?
Given: Total rooms \(= 15 + 10 + 5 = 30\), favorable outcomes \(= 10\).
$$ P(E) = \frac{10}{30} $$
$$ = \frac{1}{3} $$
$$ = 0.333 $$
Answer: 0.333 (33.3% chance)
Example 4: A retailer stocks 50 types of products; 10 are perishable. If a product is selected at random, what is the probability it is non-perishable?
Given: Total products \(= 50\), favorable outcomes \(= 50 - 10 = 40\).
$$ P(E) = \frac{40}{50} $$
$$ = 0.8 $$
Answer: 0.8 (80% chance)
Example 5: A cooperative has 8 male and 12 female members. A member is selected randomly. What is the probability the member is female?
Given: Total members \(= 8 + 12 = 20\), favorable outcomes \(= 12\).
$$ P(E) = \frac{12}{20} $$
$$ = 0.6 $$
Answer: 0.6 (60% chance)
An event is a specific outcome or set of outcomes of a random experiment. Events can be classified into different types based on their nature and relationship with other events.
Probability of an event \(E\) satisfies:
$$ 0 \leq P(E) \leq 1 $$
and for complementary events:
$$ P(E) + P(E') = 1 $$
Example 1: In a bank, the probability of a customer defaulting on a loan is 0.1. What is the probability that a customer does not default?
Given: \(P(\text{default}) = 0.1\)
$$ P(\text{no default}) = 1 - P(\text{default}) $$
$$ = 1 - 0.1 $$
$$ = 0.9 $$
Answer: 0.9 (90% chance customer does not default)
Example 2: A retail store sells three product categories: electronics, clothing, and food. The probability that a customer buys electronics is 0.3, and clothing is 0.5. If these events are mutually exclusive, what is the probability a customer buys either electronics or clothing?
Given: \(P(E) = 0.3\), \(P(C) = 0.5\), mutually exclusive.
$$ P(E \cup C) = P(E) + P(C) $$
$$ = 0.3 + 0.5 $$
$$ = 0.8 $$
Answer: 0.8 (80% chance customer buys electronics or clothing)
Example 3: In a county office, the probability that an employee attends a training session on Monday is 0.6 and on Tuesday is 0.5. If attendance on these days is independent, what is the probability that an employee attends both days?
Given: \(P(M) = 0.6\), \(P(T) = 0.5\), independent events.
$$ P(M \cap T) = P(M) \times P(T) $$
$$ = 0.6 \times 0.5 $$
$$ = 0.3 $$
Answer: 0.3 (30% chance employee attends both days)
Example 4: An agricultural cooperative notes that the probability of good rainfall in a season is 0.7. What is the probability of poor rainfall?
Given: \(P(\text{good rainfall}) = 0.7\)
$$ P(\text{poor rainfall}) = 1 - 0.7 $$
$$ = 0.3 $$
Answer: 0.3 (30% chance of poor rainfall)
Example 5: In a university, the probability a student passes mathematics is 0.85. What is the probability the student fails mathematics?
Given: \(P(\text{pass}) = 0.85\)
$$ P(\text{fail}) = 1 - 0.85 $$
$$ = 0.15 $$
Answer: 0.15 (15% chance student fails mathematics)
An outcome is a possible result of a random experiment, while the sample space \(S\) is the set of all possible outcomes. The sample space can be finite or infinite depending on the experiment. The probability of the sample space is always 1 because one of the possible outcomes must occur.
$$ S = \{1, 2, 3, 4, 5, 6\} $$
Example 1: A SACCO randomly selects one member from 5 male and 7 female members. What is the sample space and number of outcomes?
Given: Members = 5 males + 7 females
Sample space:
$$ S = \{\text{Male 1}, \text{Male 2}, \ldots, \text{Male 5}, \text{Female 1}, \ldots, \text{Female 7}\} $$
Number of outcomes:
$$ 5 + 7 = 12 $$
Answer: Sample space has 12 outcomes
Example 2: A retail shop sells 4 brands of sugar. A customer selects one brand at random. What is the sample space?
Given: 4 brands
$$ S = \{\text{Brand A}, \text{Brand B}, \text{Brand C}, \text{Brand D}\} $$
Number of outcomes:
$$ 4 $$
Answer: Sample space has 4 outcomes
Example 3: A hotel offers room types: single, double, suite. A guest checks in randomly selecting a room type. What is the sample space?
Given: 3 room types
$$ S = \{\text{Single}, \text{Double}, \text{Suite}\} $$
Number of outcomes:
$$ 3 $$
Answer: Sample space has 3 outcomes
Example 4: A county government issues vehicle permits with plate numbers ending in digits 0 to 9. What is the sample space for the last digit?
Given: digits 0 to 9
$$ S = \{0,1,2,3,4,5,6,7,8,9\} $$
Number of outcomes:
$$ 10 $$
Answer: Sample space has 10 outcomes
Example 5: A cooperative has 3 types of loans: personal, business, agricultural. A member applies for one type at random. What is the sample space?
Given: 3 loan types
$$ S = \{\text{Personal}, \text{Business}, \text{Agricultural}\} $$
Number of outcomes:
$$ 3 $$
Answer: Sample space has 3 outcomes
Simple events consist of a single outcome while compound events combine two or more simple events. The probability of compound events depends on whether the events are mutually exclusive or independent.
For mutually exclusive events \(A\) and \(B\):
$$ P(A \cup B) = P(A) + P(B) $$
For independent events \(A\) and \(B\):
$$ P(A \cap B) = P(A) \times P(B) $$
Example 1: In a retail store, the probability a customer buys bread is 0.4, and the probability of buying milk is 0.3. If these are mutually exclusive, what is the probability a customer buys bread or milk?
Given: \(P(B) = 0.4\), \(P(M) = 0.3\), mutually exclusive.
$$ P(B \cup M) = P(B) + P(M) $$
$$ = 0.4 + 0.3 $$
$$ = 0.7 $$
Answer: 0.7 (70% chance)
Example 2: A bank offers two independent loan products: personal and business. Probability a client applies for personal loan is 0.5, and for business loan is 0.2. What is the probability a client applies for both?
Given: \(P(P) = 0.5\), \(P(B) = 0.2\), independent.
$$ P(P \cap B) = P(P) \times P(B) $$
$$ = 0.5 \times 0.2 $$
$$ = 0.1 $$
Answer: 0.1 (10% chance)
Example 3: A county hospital has 8 doctors and 12 nurses. Probability a staff member is a doctor is \( \frac{8}{20} = 0.4 \), and a nurse is \(0.6\). What is the probability a randomly selected staff member is either a doctor or a nurse?
Given: mutually exclusive categories.
$$ P(\text{doctor or nurse}) = P(\text{doctor}) + P(\text{nurse}) $$
$$ = 0.4 + 0.6 $$
$$ = 1 $$
Answer: 1 (100% chance)
Example 4: In a cooperative, the probability that a member deposits money on Monday is 0.7 and on Tuesday is 0.5. If deposit days are independent, what is the probability the member deposits on both days?
Given: \(P(M) = 0.7\), \(P(T) = 0.5\), independent.
$$ P(M \cap T) = 0.7 \times 0.5 $$
$$ = 0.35 $$
Answer: 0.35 (35% chance)
Example 5: A school has 100 students: 60 boys and 40 girls. Probability a student is a boy is 0.6, a girl is 0.4. What is the probability a student is either a boy or a girl?
Given: mutually exclusive.
$$ P(\text{boy or girl}) = 0.6 + 0.4 $$
$$ = 1 $$
Answer: 1 (100% chance)
A retail business stocks 120 products, 30 of which are electronics. What is the probability a randomly selected product is electronics? (3 marks)
In a county government office, 15 employees are accountants and 25 are administrative staff. If one employee is selected at random, what is the probability the employee is an accountant or administrative staff? (3 marks)
A hotel has 50 single rooms and 20 suites. What is the probability a guest randomly assigned a room gets a suite? (3 marks)
The probability of rain in a season is 0.3. What is the probability it does not rain? (3 marks)
In a cooperative, the probability a member deposits money on Monday is 0.8, and on Tuesday is 0.6. If deposits on these days are independent, what is the probability the member deposits money on both days? (4 marks)
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Create a free accountThis chapter introduced the fundamental concepts of probability, including definitions of probability, events, outcomes, and the sample space that contains all possible outcomes. It then explored different types of events such as simple events involving a single outcome and compound events consisting of multiple outcomes, as well as independent events where the occurrence of one does not affect the other and mutually exclusive events that cannot occur simultaneously. The discussion extended to mutually inclusive events that can occur together and dependent events where the outcome of one influences the other. Key probability rules were presented, focusing on the additive rule for mutually exclusive events and the multiplicative rule for independent events. Bayes' Theorem was explained as a method to update probabilities based on new information. Finally, the chapter covered elementary probability trees as a visual tool to systematically calculate probabilities of combined events.
A retail business has 4 types of fruit: mangoes, oranges, bananas, and apples. If a customer picks one fruit at random, what is the probability that the fruit is an orange? (2 marks)
At a county referral hospital, the probability that a patient has diabetes is 0.15 and the probability that a patient has hypertension is 0.20. If these two conditions are mutually exclusive, what is the probability that a patient has either diabetes or hypertension? (3 marks)
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