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Compiled by: Wesley Rotich
Probability theory is a vital mathematical tool for decision making and risk assessment across all professional sectors in Kenya. Whether managing patient outcomes in county referral hospitals, predicting customer demand in retail businesses, or assessing compliance risks in county government offices, understanding basic probability enables professionals to make informed, data-driven choices. This chapter introduces the fundamental concepts of probability, including events, outcomes, and sample spaces, forming the foundation for more advanced statistical methods used in diverse fields such as healthcare, agriculture, finance, and environmental management.
The concept of probability quantifies the likelihood of an event occurring out of all possible outcomes. It is essential for professionals in Kenya to apply this concept when dealing with uncertainties in their daily functions, such as forecasting sales in a SACCO or evaluating the chances of equipment failure in a hotel. The following subtopics systematically develop your understanding of probability basics with practical worked examples.
Probability measures how likely an event is to happen, expressed as a number between 0 and 1, where 0 means impossible and 1 means certain. The probability of an event \(E\) occurring is defined by the ratio of the number of favorable outcomes \(n(E)\) to the total number of possible outcomes \(n(S)\) in the sample space \(S\).
$$ P(E) = \frac{n(E)}{n(S)} $$
Example 1: A county government office has 5 printers, 2 of which are faulty. What is the probability of selecting a faulty printer at random?
Given: \(n(E) = 2\) faulty printers, \(n(S) = 5\) total printers
$$ P(\text{faulty}) = \frac{2}{5} $$
$$ P(\text{faulty}) = 0.4 $$
Answer: 0.4 (or 40%)
Example 2: A retail business stocks 10 brands of cooking oil. If 3 brands are on promotion, what is the probability that a randomly chosen brand is on promotion?
Given: \(n(E) = 3\), \(n(S) = 10\)
$$ P(\text{promotion}) = \frac{3}{10} $$
$$ P(\text{promotion}) = 0.3 $$
Answer: 0.3 (or 30%)
Example 3: At a cooperative, 8 out of 40 members are farmers who grow maize. If one member is chosen at random, what is the probability that the member is a maize farmer?
Given: \(n(E) = 8\), \(n(S) = 40\)
$$ P(\text{maize farmer}) = \frac{8}{40} $$
$$ P(\text{maize farmer}) = 0.2 $$
Answer: 0.2 (or 20%)
Example 4: A hotel has 12 rooms, 4 of which are booked. If a room is selected at random, what is the probability that it is not booked?
Given: \(n(E) = 12 - 4 = 8\) available rooms, \(n(S) = 12\)
$$ P(\text{not booked}) = \frac{8}{12} $$
$$ P(\text{not booked}) = \frac{2}{3} = 0.6667 $$
Answer: 0.6667 (or 66.67%)
Example 5: In a sample of 15 environmental reports submitted to NEMA, 5 were approved. What is the probability that a randomly selected report is approved?
Given: \(n(E) = 5\), \(n(S) = 15\)
$$ P(\text{approved}) = \frac{5}{15} $$
$$ P(\text{approved}) = \frac{1}{3} = 0.3333 $$
Answer: 0.3333 (or 33.33%)
An event is a specific outcome or a set of outcomes from a random experiment. Events are classified into simple and compound, mutually exclusive or non-mutually exclusive, and independent or dependent, which influence how their probabilities are calculated.
Example 1: In a bank, the probability of a client defaulting on a loan is 0.1, and the probability of a client requesting a loan extension is 0.2. Assuming these events are independent, what is the probability that a client both defaults and requests an extension?
Given: \(P(D) = 0.1\), \(P(E) = 0.2\), independent events
$$ P(D \text{ and } E) = P(D) \times P(E) $$
$$ P(D \text{ and } E) = 0.1 \times 0.2 $$
$$ P(D \text{ and } E) = 0.02 $$
Answer: 0.02 (or 2%)
Example 2: At a TVET college, 30% of students study ICT and 40% study business. If 20% study both ICT and business, what is the probability a student studies ICT or business?
Given: \(P(ICT) = 0.3\), \(P(B) = 0.4\), \(P(ICT \cap B) = 0.2\)
$$ P(ICT \cup B) = P(ICT) + P(B) - P(ICT \cap B) $$
$$ P(ICT \cup B) = 0.3 + 0.4 - 0.2 $$
$$ P(ICT \cup B) = 0.5 $$
Answer: 0.5 (or 50%)
Example 3: A SACCO has 100 members. 60 members save regularly, and 45 members attend monthly meetings. If 30 members both save regularly and attend meetings, calculate the probability that a randomly selected member either saves regularly or attends meetings.
Given: \(n(S) = 100\), \(n(Save) = 60\), \(n(Meet) = 45\), \(n(Save \cap Meet) = 30\)
$$ P(Save \cup Meet) = \frac{60}{100} + \frac{45}{100} - \frac{30}{100} $$
$$ P(Save \cup Meet) = 0.6 + 0.45 - 0.3 $$
$$ P(Save \cup Meet) = 0.75 $$
Answer: 0.75 (or 75%)
Example 4: In a farm cooperative, the probability that a farmer uses fertilizer is 0.7, and the probability that a farmer uses irrigation is 0.5. If these two events are mutually exclusive, what is the probability that a farmer uses either fertilizer or irrigation?
Given: \(P(F) = 0.7\), \(P(I) = 0.5\), mutually exclusive events
$$ P(F \cup I) = P(F) + P(I) $$
$$ P(F \cup I) = 0.7 + 0.5 $$
Since probability cannot exceed 1, this indicates an inconsistency; mutually exclusive events must satisfy \(P(F) + P(I) \leq 1\).
Answer: The sum exceeds 1, meaning the assumption of mutual exclusivity is incorrect.
Example 5: At a county referral hospital, the probability that a patient has condition A is 0.15, condition B is 0.25, and both conditions occur together with probability 0.05. Find the probability that a patient has condition A or B.
Given: \(P(A) = 0.15\), \(P(B) = 0.25\), \(P(A \cap B) = 0.05\)
$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$
$$ P(A \cup B) = 0.15 + 0.25 - 0.05 $$
$$ P(A \cup B) = 0.35 $$
Answer: 0.35 (or 35%)
An outcome is a possible result of an experiment or trial. Enumerating all possible outcomes is crucial for calculating probabilities. Techniques such as listing, tree diagrams, and the multiplication principle help organize outcomes, especially when dealing with multiple stages or events.
Example 1: A retail shop sells 3 types of fruit: apples, bananas, and oranges. If a customer buys one fruit, how many possible outcomes are there?
Given: 3 fruit types
Number of outcomes = 3
Answer: 3 possible outcomes
Example 2: A SACCO offers 2 types of savings accounts and 3 types of loan products. How many different product combinations can a client choose if they select one savings account and one loan product?
Given: 2 savings types, 3 loan types
$$ \text{Total outcomes} = 2 \times 3 = 6 $$
Answer: 6 product combinations
Example 3: A hotel offers breakfast choices of tea or coffee and main dishes of eggs, pancakes, or porridge. How many breakfast combinations are possible?
Given: 2 drink options, 3 dishes
$$ \text{Total outcomes} = 2 \times 3 = 6 $$
Answer: 6 breakfast combinations
Example 4: In a county government office, a new employee ID consists of 2 letters followed by 3 digits. How many different IDs can be created if letters and digits can repeat?
Given: 26 letters, 10 digits
$$ \text{Total outcomes} = 26^2 \times 10^3 $$
$$ = 676 \times 1000 $$
$$ = 676,000 $$
Answer: 676,000 unique IDs
Example 5: A farmer plants 4 varieties of maize and 3 varieties of beans. How many different pairs of maize and bean varieties can be planted together?
Given: 4 maize varieties, 3 bean varieties
$$ \text{Total outcomes} = 4 \times 3 = 12 $$
Answer: 12 planting pairs
The sample space \(S\) is the set of all possible outcomes of an experiment. Identifying the sample space accurately is fundamental for calculating probabilities. Sample spaces can be finite or infinite, discrete or continuous, and are often represented using sets, lists, or diagrams.
Example 1: What is the sample space when rolling a standard six-sided die?
Sample space:
$$ S = \{1, 2, 3, 4, 5, 6\} $$
Number of outcomes: 6
Answer: \(S = \{1, 2, 3, 4, 5, 6\}\)
Example 2: What is the sample space when tossing two coins simultaneously?
Sample space:
$$ S = \{HH, HT, TH, TT\} $$
Number of outcomes: 4
Answer: \(S = \{HH, HT, TH, TT\}\)
Example 3: A university offers 3 courses: Accounting, Marketing, and IT. A student chooses one course. What is the sample space?
Sample space:
$$ S = \{\text{Accounting}, \text{Marketing}, \text{IT}\} $$
Number of outcomes: 3
Answer: \(S = \{\text{Accounting}, \text{Marketing}, \text{IT}\}\)
Example 4: A farmer plants different seed types: maize, beans, sorghum. If the farmer selects one seed type at random, find the sample space.
Sample space:
$$ S = \{\text{maize}, \text{beans}, \text{sorghum}\} $$
Number of outcomes: 3
Answer: \(S = \{\text{maize}, \text{beans}, \text{sorghum}\}\)
Example 5: A SACCO issues membership cards numbered from 001 to 100. What is the sample space?
Sample space:
$$ S = \{001, 002, 003, ..., 100\} $$
Number of outcomes: 100
Answer: \(S = \{001, 002, ..., 100\}\)
A hospital has 8 ambulances, 3 of which are currently in use. What is the probability that a randomly selected ambulance is available? (3 marks)
A hotel offers 4 types of breakfast and 5 types of lunch. How many meal combinations can a guest choose if they order one breakfast and one lunch? (3 marks)
At a retail store, 20% of the products are electronic devices, and 15% are kitchen appliances. If 5% are both, what is the probability that a product is electronic or a kitchen appliance? (4 marks)
A cooperative has 50 members, 30 are farmers, 20 are traders, and 10 are both farmers and traders. What is the probability a member is either a farmer or a trader? (4 marks)
A county government office assigns employee codes starting with one letter (A-Z) followed by two digits (00-99). How many unique codes can be generated? (4 marks)
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Create a free accountThis chapter introduces the fundamental concepts of probability, including definitions of probability, events, outcomes, and the sample space. It explains different types of events such as simple and compound events, as well as independent and mutually exclusive events. The chapter also covers mutually inclusive and dependent events, highlighting their relationships and distinctions. Key rules of probability are presented, focusing on the additive and multiplicative rules for calculating probabilities of combined events. Bayes' Theorem is explored, providing a method to update probabilities based on new information. Finally, elementary probability trees are introduced as a visual tool to systematically represent and solve probability problems involving sequential events. Together, these topics provide a comprehensive foundation for understanding and applying basic probability theory in business mathematics and statistics.
A SACCO has 5 members who apply for a loan. Each member has an equal chance of being approved. What is the probability that a randomly selected member is approved? (1 mark)
At a county referral hospital, 3 types of tests (A, B, and C) are conducted. The probability of a patient needing test A is 0.4, test B is 0.3, and test C is 0.2. What is the probability that a patient needs either test A or test B? (2 marks)
A county hospital laboratory performs tests on patient samples. If the probability of a sample testing positive for a certain infection is \(0.15\), what is the probability that a randomly selected sample tests negative? (4 marks)
In a retail business, there are 3 types of products: A, B, and C. If a customer buys product A with probability \(0.4\) and product B with probability \(0.3\), assuming these events are mutually exclusive, what is the probability that the customer buys either product A or product B? (4 marks)
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