By the end of this chapter, you will be able to:
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Probability theory is a fundamental tool in accountancy for assessing risks and uncertainties in financial decisions. It enables accountants to quantify the likelihood of various outcomes, which is crucial for auditing, forecasting, and investment analysis. In Kenya’s dynamic business environment, understanding probability helps professionals in banks, SACCOs, and retail businesses to make informed decisions under uncertainty, improving accuracy in financial reports and risk management.
Probability theory begins with understanding the basic concepts that describe the chance of occurrence of different events. These include outcomes, which are possible results of an experiment; events, which are sets of outcomes; and the sample space, which is the set of all possible outcomes. Mastery of these concepts is essential for accountants to apply probability in areas such as credit risk assessment, inventory management, and financial forecasting.
Probability measures how likely an event is to occur, expressed as a number between 0 (impossible event) and 1 (certain event). The outcome is a single possible result of a random experiment, such as rolling a die or drawing a card. The sample space (denoted \( S \)) is the set of all possible outcomes of the experiment. An event is any subset of the sample space, meaning it can include one or more outcomes.
The probability of an event \( E \) is written as \( P(E) \). For any event, the following must hold:
The classical definition of probability for equally likely outcomes is:
$$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes in } S} $$
Example 1:
A SACCO decides to randomly select one member from 10 applicants for a loan. What is the probability that a particular member is selected?
Given: Number of applicants = 10, favorable outcomes = 1$$ P(E) = \frac{1}{10} $$$$ P(E) = 0.1 $$
Answer: 0.1 (10%)
Example 2:
A retail store manager randomly picks one item from 20 items on sale. If 5 of these are defective, what is the probability of picking a defective item?
Given: Total items = 20, defective items = 5$$ P(E) = \frac{5}{20} $$$$ P(E) = \frac{1}{4} $$$$ P(E) = 0.25 $$
Answer: 0.25 (25%)
Example 3:
A financial auditor selects one transaction at random from 100 transactions. If 2 transactions are known to be fraudulent, find the probability that the auditor selects a fraudulent transaction.
Given: Total transactions = 100, fraudulent = 2$$ P(E) = \frac{2}{100} $$$$ P(E) = 0.02 $$
Answer: 0.02 (2%)
Example 4:
A bank has 8 loan applications, 3 of which are approved. If one application is randomly selected, what is the probability it is approved?
Given: Total applications = 8, approved = 3$$ P(E) = \frac{3}{8} $$$$ P(E) = 0.375 $$
Answer: 0.375 (37.5%)
Example 5:
A company has 12 employees; 4 are accountants, 5 are marketers, and 3 are IT staff. If one employee is randomly chosen, what is the probability that the employee is either an accountant or marketer?
Given: Accountants = 4, Marketers = 5, Total employees = 12$$ P(E) = \frac{4 + 5}{12} $$$$ P(E) = \frac{9}{12} $$$$ P(E) = 0.75 $$
Answer: 0.75 (75%)
Events in probability can be classified into several types important for analyzing financial and business situations. The key types are:
The probability rules for these types are essential in calculating combined probabilities in accounting tasks such as fraud detection or financial risk evaluation.
Example 1:
In a bank, the probability that a customer defaults on a loan is 0.1. What is the probability that the customer does not default?
Given: \( P(\text{default}) = 0.1 \)$$ P(\text{not default}) = 1 - P(\text{default}) $$$$ P(\text{not default}) = 1 - 0.1 $$$$ P(\text{not default}) = 0.9 $$
Answer: 0.9 (90%)
Example 2:
A SACCO has 100 members: 40 are farmers, 30 are traders, and 30 are teachers. If a member is randomly selected, what is the probability that the member is either a farmer or a trader?
Given: Farmers = 40, Traders = 30, Total = 100$$ P(\text{farmer or trader}) = P(\text{farmer}) + P(\text{trader}) $$$$ P(\text{farmer or trader}) = \frac{40}{100} + \frac{30}{100} $$$$ P(\text{farmer or trader}) = 0.4 + 0.3 $$$$ P(\text{farmer or trader}) = 0.7 $$
Answer: 0.7 (70%)
Example 3:
In an audit, the probability of detecting an error in one transaction is 0.05. If two transactions are audited independently, what is the probability that errors are detected in both?
Given: \( P(\text{error in one}) = 0.05 \), independent events$$ P(\text{error in both}) = P(\text{error in first}) \times P(\text{error in second}) $$$$ P(\text{error in both}) = 0.05 \times 0.05 $$$$ P(\text{error in both}) = 0.0025 $$
Answer: 0.0025 (0.25%)
Example 4:
A retailer checks two suppliers. The probability that supplier A delivers on time is 0.8, and supplier B delivers on time is 0.9. Assuming independence, what is the probability that both suppliers deliver on time?
Given: \( P(A) = 0.8 \), \( P(B) = 0.9 \)$$ P(\text{both on time}) = P(A) \times P(B) $$$$ P(\text{both on time}) = 0.8 \times 0.9 $$$$ P(\text{both on time}) = 0.72 $$
Answer: 0.72 (72%)
Example 5:
In a financial report, the probability that a particular account is error-free is 0.95. What is the probability that it contains an error?
Given: \( P(\text{error-free}) = 0.95 \)$$ P(\text{error}) = 1 - 0.95 $$$$ P(\text{error}) = 0.05 $$
Answer: 0.05 (5%)
The sample space \( S \) is the set of all possible outcomes of an experiment. Representing the sample space correctly is crucial for calculating probabilities accurately. Sample spaces can be finite or infinite, but in accountancy applications, they are usually finite and discrete.
Common ways to represent sample spaces include:
Example 1:
A SACCO randomly selects one member from 3 applicants: Alice, Brian, and Carol. Write the sample space.
Given: Applicants = {Alice, Brian, Carol}
Sample space:$$ S = \{\text{Alice}, \text{Brian}, \text{Carol}\} $$
Answer: \( S = \{\text{Alice}, \text{Brian}, \text{Carol}\} \)
Example 2:
A financial auditor flips a coin and rolls a 6-sided die. List the sample space of all outcomes.
Given: Coin outcomes = {Heads, Tails}, Die outcomes = {1, 2, 3, 4, 5, 6}
Sample space:$$ S = \{(H,1), (H,2), (H,3), (H,4), (H,5), (H,6), (T,1), (T,2), (T,3), (T,4), (T,5), (T,6)\} $$
Answer: 12 outcomes as above
Example 3:
A bank records the day of the week a loan application is submitted (Monday to Friday). Write the sample space.
Given: Days = Monday, Tuesday, Wednesday, Thursday, Friday$$ S = \{\text{Mon}, \text{Tue}, \text{Wed}, \text{Thu}, \text{Fri}\} $$
Answer: \( S = \{\text{Mon}, \text{Tue}, \text{Wed}, \text{Thu}, \text{Fri}\} \)
Example 4:
A SACCO member’s loan application can be approved, rejected, or pending. Write the sample space.
Given: Possible results = Approve, Reject, Pending$$ S = \{\text{Approve}, \text{Reject}, \text{Pending}\} $$
Answer: \( S = \{\text{Approve}, \text{Reject}, \text{Pending}\} \)
Example 5:
A retailer tests two products, each can be either defective (D) or non-defective (N). Write the sample space.
Given: Product 1 = {D, N}, Product 2 = {D, N}
Sample space:$$ S = \{(D,D), (D,N), (N,D), (N,N)\} $$
Answer: 4 outcomes as above
An event \( E \) is any subset of the sample space. Events can be simple (one outcome) or compound (multiple outcomes). The probability of an event is the sum of the probabilities of the outcomes it contains.
For equally likely outcomes:
$$ P(E) = \frac{|E|}{|S|} $$
where \( |E| \) is the number of favorable outcomes and \( |S| \) is the total number of outcomes.
For events with unequal probabilities, the total probability is the sum of individual outcome probabilities.
Example 1:
A SACCO has 5 applicants: 2 are women, 3 are men. If one applicant is randomly selected, find the probability of selecting a woman.
Given: Women = 2, Total = 5$$ P(E) = \frac{2}{5} $$$$ P(E) = 0.4 $$
Answer: 0.4 (40%)
Example 2:
A retailer has 4 products: 1 defective, 3 non-defective. What is the probability that a product picked at random is non-defective?
Given: Non-defective = 3, Total = 4$$ P(E) = \frac{3}{4} $$$$ P(E) = 0.75 $$
Answer: 0.75 (75%)
Example 3:
A bank’s loan approval outcomes for a day are: 3 approved, 2 rejected, 1 pending. What is the probability that a randomly selected outcome is either approved or pending?
Given: Approved = 3, Pending = 1, Total = 6$$ P(E) = \frac{3 + 1}{6} $$$$ P(E) = \frac{4}{6} $$$$ P(E) = \frac{2}{3} $$$$ P(E) \approx 0.6667 $$
Answer: 0.6667 (66.67%)
Example 4:
An auditor finds errors in 5 out of 20 transactions. What is the probability that a transaction randomly selected contains an error?
Given: Errors = 5, Total = 20$$ P(E) = \frac{5}{20} $$$$ P(E) = 0.25 $$
Answer: 0.25 (25%)
Example 5:
A SACCO has 10 members, 4 have defaulted on loans. What is the probability that a member selected at random has not defaulted?
Given: Defaulted = 4, Not defaulted = 6, Total = 10$$ P(E) = \frac{6}{10} $$$$ P(E) = 0.6 $$
Answer: 0.6 (60%)
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Create a free accountThis chapter introduced the fundamental concepts of probability, including key definitions such as events, outcomes, and sample space, which form the basis of understanding probability theory. It explored various types of events, distinguishing between simple and compound events, as well as independent, mutually exclusive, mutually inclusive, and dependent events, emphasizing their roles in probability calculations and report writing. The chapter then covered essential probability rules, focusing on the additive rule for mutually exclusive events and the multiplicative rule for independent events, providing methods to calculate combined probabilities accurately. Bayes' Theorem was presented as a powerful tool for updating probabilities based on new information, enhancing decision-making under uncertainty. Finally, the chapter explained elementary probability trees as visual aids that simplify the analysis of complex probability problems by mapping possible outcomes and their probabilities systematically. Together, these topics equip students with a comprehensive foundation to apply basic probability concepts effectively in various business and statistical contexts.
A company has 5 different invoices to pay. What is the probability that when selecting one invoice at random, it is the invoice for Ksh 10,000? (1 mark)
In an audit, the probability that a randomly chosen transaction is an expense is 0.6. What is the probability that the transaction is not an expense? (1 mark)
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