Probability Basics

Learning Objective: Calculate probabilities for simple business scenarios

A Nepali insurance company needs to estimate the chance that a 30-year-old policyholder will make a health claim this year. A factory manager wants to know the probability that a machine will break down during peak production. Probability helps businesses quantify uncertainty and make better decisions.

What Is Probability?

Probability is a measure of how likely an event is to occur. It ranges from 0 (impossible) to 1 (certain).

Formula: P(Event) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Example: When rolling a fair die, the probability of getting a 4 is: P(4) = 1/6 = 0.167 or 16.7%

Key Terms

• Experiment: An action with uncertain outcomes (e.g., tossing a coin)
• Sample Space (S): The set of all possible outcomes. For a coin toss: S = {Head, Tail}
• Event: A specific outcome or set of outcomes (e.g., getting Heads)
• Complementary Event: The opposite of an event. P(not A) = 1 - P(A)

Basic Probability Rules

Addition Rule (OR)

Used when we want the probability of either event A or event B occurring.

For mutually exclusive events (cannot happen together): P(A or B) = P(A) + P(B)

Example: A bag has 5 red balls and 3 blue balls. What is the probability of picking a red OR blue ball? P(Red or Blue) = 5/8 + 3/8 = 8/8 = 1 (certain, since those are all the balls)

For non-mutually exclusive events: P(A or B) = P(A) + P(B) - P(A and B)

Example: In a class of 40 students, 25 study accounting and 20 study economics. 10 study both. What is the probability a randomly chosen student studies accounting OR economics? P(A or E) = 25/40 + 20/40 - 10/40 = 35/40 = 7/8 = 0.875

Multiplication Rule (AND)

Used when we want the probability of event A and event B both occurring.

For independent events (one does not affect the other): P(A and B) = P(A) x P(B)

Example: A Nepali company has two machines. Machine A has a 10% chance of breaking down on any day, and Machine B has a 5% chance. What is the probability both break down on the same day? P(A and B) = 0.10 x 0.05 = 0.005 or 0.5%

Tree Diagrams

Tree diagrams visually map out all possible outcomes of sequential events.

Business Example: A quality control inspector at a garment factory checks shirts. 90% of shirts pass quality control. Of those that fail, 60% can be repaired.

                Start
               /     \
          Pass(0.9)  Fail(0.1)
                     /       \
              Repair(0.06)  Reject(0.04)

• P(Pass) = 0.9
• P(Fail and Repair) = 0.1 x 0.6 = 0.06
• P(Fail and Reject) = 0.1 x 0.4 = 0.04

Business Applications of Probability

• Insurance: Calculating premiums based on probability of claims
• Quality control: Estimating defect rates in production
• Marketing: Predicting the chance a customer will buy after seeing an ad
• Finance: Assessing risk of loan defaults

Key Term: Mutually exclusive events are events that cannot occur at the same time. If you flip a coin, you cannot get both heads and tails simultaneously.

Summary

• Probability measures the likelihood of an event, ranging from 0 to 1.
• The Addition Rule calculates P(A or B); the Multiplication Rule calculates P(A and B).
• Tree diagrams help visualize sequential events and their probabilities.
• Businesses use probability for insurance pricing, quality control, and risk assessment.