Probability is an essential concept for industrial control and installation engineers in Kenya, as it enables them to model uncertainty and make informed decisions about system reliability, maintenance scheduling, and risk management. Whether designing backup systems for a manufacturing plant, analyzing the likelihood of equipment failure, or planning preventive maintenance, probability theory provides the tools to quantify and manage risks in complex engineering environments. Understanding probability and its calculations is vital for optimizing resource allocation, minimizing downtime, and ensuring safe, efficient operations in Kenyan industries.

4.1 Probability

Probability quantifies the likelihood of events occurring, such as the failure of a component in a control system or the successful completion of an installation within a given timeframe. Engineers use probability to assess risks, plan maintenance, and design reliable systems.

4.1.1 Laws of Probability

The laws of probability govern how probabilities are calculated and combined for different types of events. The two fundamental laws are the Addition Law (for mutually exclusive events) and the Multiplication Law (for independent events).

Addition Law

If events \(A\) and \(B\) are mutually exclusive (cannot happen at the same time), the probability that either occurs is:$$ P(A \text{ or } B) = P(A) + P(B) $$

Multiplication Law

If events \(A\) and \(B\) are independent (the occurrence of one does not affect the other), the probability that both occur is:$$ P(A \text{ and } B) = P(A) \times P(B) $$

Worked Examples

Example 1 (Easy):
A technician is installing two independent control modules. The probability that Module A fails during installation is \(0.1\), and Module B fails is \(0.2\). What is the probability that both fail?

Given:
\(P(A) = 0.1\)
\(P(B) = 0.2\)

$$ P(A \text{ and } B) = P(A) \times P(B) $$$$ = 0.1 \times 0.2 $$$$ = 0.02 $$

Answer: \(0.02\) or 2%

Example 2 (Medium):
A maintenance planner knows the probability of a power surge (Event A) is \(0.05\), and the probability of a water leak (Event B) is \(0.03\). If these events are mutually exclusive, what is the probability that either occurs during a week?

Given:
\(P(A) = 0.05\)
\(P(B) = 0.03\)

$$ P(A \text{ or } B) = P(A) + P(B) $$$$ = 0.05 + 0.03 $$$$ = 0.08 $$

Answer: \(0.08\) or 8%

Example 3 (Hard):
At a county government water treatment facility, the probability that a pump fails in a month is \(0.04\), and the probability that a sensor fails is \(0.06\). If these events are independent, what is the probability that both fail in the same month?

Given:
\(P(\text{Pump fails}) = 0.04\)
\(P(\text{Sensor fails}) = 0.06\)

$$ P(\text{Both fail}) = 0.04 \times 0.06 $$$$ = 0.0024 $$

Answer: \(0.0024\) or 0.24%

Example 4 (Applied):
In a Nairobi hotel, the probability that the backup generator is needed during a power outage is \(0.15\), and the probability that the generator fails when needed is \(0.02\). What is the probability that both events occur?

Given:
\(P(\text{Generator needed}) = 0.15\)
\(P(\text{Generator fails}) = 0.02\)

$$ P(\text{Both}) = 0.15 \times 0.02 $$$$ = 0.003 $$

Answer: \(0.003\) or 0.3%

Example 5 (Challenge):
In a SACCO office, the probability of a fire alarm being triggered falsely is \(0.01\), and the probability of an actual fire occurring is \(0.005\). If these events are mutually exclusive, what is the probability that either event occurs?

Given:
\(P(\text{False alarm}) = 0.01\)
\(P(\text{Actual fire}) = 0.005\)

$$ P(\text{Alarm or Fire}) = 0.01 + 0.005 $$$$ = 0.015 $$

Answer: \(0.015\) or 1.5%

Chapter Summary

This chapter introduced the foundational concept of probability, emphasizing the fundamental laws that govern how probabilities are calculated and interpreted. It covered expectation, variance, and standard deviation, equipping students with tools to measure the average outcome and spread of random events. Various types of probability distributions were explored, with a focus on how to determine their mean, variance, and standard deviation. The chapter also discussed the classification of events into dependent, independent, and mutually exclusive types, highlighting how these relationships affect probability calculations. Essential counting techniques such as permutations, combinations, tree diagrams, and Venn diagrams were explained to help organize and solve complex probability problems. Finally, real-world applications of probability distributions were addressed, demonstrating their value in engineering decision-making and risk assessment.