By the end of this chapter, you will be able to:
Mastering these skills will help you solve real-world problems and make informed decisions in your trade using quantitative techniques.
Determining probability and probabilistic distribution is essential for accountancy professionals in Kenya who often face uncertainty in financial forecasting, risk assessment, and decision-making under uncertainty. Understanding probability allows accountants to quantify the likelihood of various financial outcomes, such as default rates on loans, budget variances, or market fluctuations. Probabilistic distributions help model these uncertainties to improve accuracy in financial reports and strategic planning. This chapter develops foundational skills to calculate probabilities and apply probabilistic models relevant to accountancy tasks.
This section introduces key concepts and terms related to probability and probabilistic distributions, providing the basis for quantitative analysis in accountancy contexts.
Probability measures the chance that a particular event will occur out of all possible outcomes. It quantifies uncertainty and ranges between 0 (impossible event) and 1 (certain event). Accountants use probability to evaluate risks such as loan defaults or investment returns.
The probability \(P\) of an event \(E\) is given by:
$$ P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} $$
Example 1: A bank has 5 loan applicants, 2 of whom are likely to default. What is the probability that a randomly selected applicant will default?
Given: Number of favourable outcomes = 2, total outcomes = 5$$P(\text{default}) = \frac{2}{5}$$$$P(\text{default}) = 0.4$$ Answer: 0.4 (or 40%)
Example 2: An accountant is auditing 10 invoices, 3 of which contain errors. What is the probability of selecting an invoice with an error at random?
Given: Number of favourable outcomes = 3, total outcomes = 10$$P(\text{error}) = \frac{3}{10}$$$$P(\text{error}) = 0.3$$ Answer: 0.3 (or 30%)
Example 3: In a SACCO, there are 8 members, 5 are active savers, and 3 are inactive. What is the probability of selecting an active saver?
Given: Number of favourable outcomes = 5, total outcomes = 8$$P(\text{active saver}) = \frac{5}{8}$$$$P(\text{active saver}) = 0.625$$ Answer: 0.625 (or 62.5%)
Example 4: A county government office processes 50 payments daily, 5 are delayed. What is the probability a randomly chosen payment is not delayed?
Given: Number of favourable outcomes = 45 (not delayed), total outcomes = 50$$P(\text{not delayed}) = \frac{45}{50}$$$$P(\text{not delayed}) = 0.9$$ Answer: 0.9 (or 90%)
Example 5: A retailer stocks 20 items, 4 are defective. What is the probability of selecting a non-defective item?
Given: Number of favourable outcomes = 16, total outcomes = 20$$P(\text{non-defective}) = \frac{16}{20}$$$$P(\text{non-defective}) = 0.8$$ Answer: 0.8 (or 80%)
The complement of an event \(E\) is the event that \(E\) does not occur. It is denoted as \(E^c\). The sum of the probabilities of an event and its complement is always 1. This concept is important when calculating the likelihood of an event not happening, such as the probability of a loan not defaulting.
The formula is:
$$ P(E^c) = 1 - P(E) $$
Example 1: If the probability of default on a loan is 0.25, find the probability that the loan will not default.
Given: \(P(E) = 0.25\)$$P(E^c) = 1 - 0.25$$$$P(E^c) = 0.75$$ Answer: 0.75 (or 75%)
Example 2: An invoice has a 0.1 probability of containing an error. What is the probability the invoice is error-free?
Given: \(P(E) = 0.1\)$$P(E^c) = 1 - 0.1$$$$P(E^c) = 0.9$$ Answer: 0.9 (or 90%)
Example 3: The probability of a payment being delayed is 0.2. Find the probability it is processed on time.
Given: \(P(E) = 0.2\)$$P(E^c) = 1 - 0.2$$$$P(E^c) = 0.8$$ Answer: 0.8 (or 80%)
Example 4: A SACCO has a 0.15 probability that a member defaults on a loan. Find the probability that a member does not default.
Given: \(P(E) = 0.15\)$$P(E^c) = 1 - 0.15$$$$P(E^c) = 0.85$$ Answer: 0.85 (or 85%)
Example 5: The probability that an audit reveals a discrepancy is 0.05. Find the probability that no discrepancy is found.
Given: \(P(E) = 0.05\)$$P(E^c) = 1 - 0.05$$$$P(E^c) = 0.95$$ Answer: 0.95 (or 95%)
Mutually exclusive events cannot happen at the same time. For example, an invoice cannot both be approved and rejected simultaneously. Non-mutually exclusive events can occur together, such as a client being both a borrower and a saver in a bank.
For mutually exclusive events \(A\) and \(B\):
$$ P(A \cup B) = P(A) + P(B) $$
For non-mutually exclusive events:
$$ P(A \cup B) = P(A) + P(B) - P(A \cap B) $$
Example 1: In a retail business, the probability that a customer buys product A is 0.4, and product B is 0.3. The two products cannot be bought together. Find the probability a customer buys product A or B.
Given: \(P(A) = 0.4\), \(P(B) = 0.3\) (mutually exclusive)$$P(A \cup B) = 0.4 + 0.3$$$$P(A \cup B) = 0.7$$ Answer: 0.7 (or 70%)
Example 2: In a bank, 40% of clients have savings accounts, 30% have loans, and 10% have both. Find the probability a client has either a savings account or a loan.
Given: \(P(A) = 0.4\), \(P(B) = 0.3\), \(P(A \cap B) = 0.1\) (non-mutually exclusive)$$P(A \cup B) = 0.4 + 0.3 - 0.1$$$$P(A \cup B) = 0.6$$ Answer: 0.6 (or 60%)
Example 3: A county government office finds that 25% of staff are accountants, 15% are auditors, and no one holds both roles. Find the probability a staff member is either an accountant or auditor.
Given: \(P(A) = 0.25\), \(P(B) = 0.15\) (mutually exclusive)$$P(A \cup B) = 0.25 + 0.15$$$$P(A \cup B) = 0.4$$ Answer: 0.4 (or 40%)
Example 4: In a cooperative, 60% of members are farmers, 50% are traders, and 30% are both. Find the probability a member is either a farmer or trader.
Given: \(P(A) = 0.6\), \(P(B) = 0.5\), \(P(A \cap B) = 0.3\) (non-mutually exclusive)$$P(A \cup B) = 0.6 + 0.5 - 0.3$$$$P(A \cup B) = 0.8$$ Answer: 0.8 (or 80%)
Example 5: In a hotel, 20% of guests pay by cash, 50% by card, and none pay by both. Find the probability a guest pays by cash or card.
Given: \(P(A) = 0.2\), \(P(B) = 0.5\) (mutually exclusive)$$P(A \cup B) = 0.2 + 0.5$$$$P(A \cup B) = 0.7$$ Answer: 0.7 (or 70%)
Independent events are those where the occurrence of one event does not affect the probability of another. Dependent events occur when one event influences the probability of the other.
For independent events \(A\) and \(B\):
$$ P(A \cap B) = P(A) \times P(B) $$
For dependent events:
$$ P(A \cap B) = P(A) \times P(B|A) $$
Where \(P(B|A)\) is the probability of \(B\) given \(A\) has occurred.
Example 1: A SACCO has a 0.3 probability a member saves this month, and a 0.4 probability the member takes a loan, independently. Find the probability a member saves and takes a loan.
Given: \(P(A) = 0.3\), \(P(B) = 0.4\) (independent)$$P(A \cap B) = 0.3 \times 0.4$$$$P(A \cap B) = 0.12$$ Answer: 0.12 (or 12%)
Example 2: In a bank, the probability a client applies for a loan is 0.5. If the client applies, the probability of approval is 0.7. Find the probability a client applies and is approved.
Given: \(P(A) = 0.5\), \(P(B|A) = 0.7\) (dependent)$$P(A \cap B) = 0.5 \times 0.7$$$$P(A \cap B) = 0.35$$ Answer: 0.35 (or 35%)
Example 3: An auditor finds that the probability of an error in accounts is 0.1. Given an error, the probability it is detected is 0.8. Find the probability an error is detected.
Given: \(P(A) = 0.1\), \(P(B|A) = 0.8\) (dependent)$$P(A \cap B) = 0.1 \times 0.8$$$$P(A \cap B) = 0.08$$ Answer: 0.08 (or 8%)
Example 4: In a retail business, 70% of customers pay by cash, and 40% of cash payers use loyalty cards. Find the probability a customer pays by cash and uses a loyalty card.
Given: \(P(A) = 0.7\), \(P(B|A) = 0.4\) (dependent)$$P(A \cap B) = 0.7 \times 0.4$$$$P(A \cap B) = 0.28$$ Answer: 0.28 (or 28%)
Example 5: The probability that a company submits financial reports on time is 0.9. If submitted on time, the probability that the report passes audit is 0.95. Find the probability reports are submitted on time and pass audit.
Given: \(P(A) = 0.9\), \(P(B|A) = 0.95\) (dependent)$$P(A \cap B) = 0.9 \times 0.95$$$$P(A \cap B) = 0.855$$ Answer: 0.855 (or 85.5%)
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Create a free accountThis chapter explored the fundamental terms related to probability and probabilistic distributions, establishing a clear understanding of key concepts such as events, outcomes, and sample spaces. It then introduced the basic principles of probability, including the calculation of likelihoods for different events. The chapter proceeded to explain the essential laws of probability distribution, highlighting rules like addition and multiplication that govern how probabilities combine. A detailed examination of probability distribution functions followed, describing how these functions model the behavior of random variables. The practical application of these distribution functions was demonstrated through the preparation and interpretation of reports. Finally, the chapter concluded with an introduction to hypothesis testing, showing how probability distributions are used to assess and make decisions based on data. This comprehensive overview equips students with the skills to apply probability theory effectively in quantitative analysis.
A company’s audit department finds that the probability of detecting an accounting error in a financial report is 0.3. What is the probability that the audit department will not detect an error? (Easy 1 mark)
In a sample of 200 invoices, the probability that an invoice contains a clerical error is 0.05. How many invoices are expected to have errors? (Easy 1 mark)
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