2 Probability for Business

Probability and business uncertainty

Probability measures uncertainty by assigning each event a value from $0$, meaning impossible, to $1$, meaning certain. Businesses use probabilities to assess risks, forecast outcomes, and compare decisions. An event is an outcome or group of outcomes, such as a customer placing an order or a shipment arriving late.

Complements and combining events

For an event $A$, its complement is the event that $A$ does not occur. The probability of the complement is

For two events $A$ and $B$, the addition rule gives the probability that at least one occurs:

The subtraction prevents outcomes belonging to both events from being counted twice. If the events are mutually exclusive, they cannot occur together, so $P(A\cap B)=0$ and the rule reduces to $P(A\cup B)=P(A)+P(B)$. These rules can be used to combine business risks or calculate the chance that at least one target is met.

For example, let $E$ mean that an order shipped express and $R$ mean that it was returned. If $P(E)=0.12$, $P(R)=0.08$, and $P(E\cap R)=0.03$, then

Thus, $17\%$ of the orders were express, returned, or both.

Conditional probability

Conditional probability is the probability of an event when another event is known to have occurred. For $P(B)>0$, its definition is

The condition after the vertical bar specifies the group or circumstances being considered. In the order example, the probability that an order was returned given that it shipped express is

Therefore, $25\%$ of express orders were returned. This conditional rate may be more useful to an operations team than the overall return rate because it focuses on the relevant order type.

The multiplication rule

Rearranging the definition of conditional probability gives the multiplication rule:

For three events, applying the rule successively gives

This chain rule does not require the events to be independent.

Independence and mutual exclusivity

Events $A$ and $B$ are independent when knowing that one occurred does not change the probability of the other. When the relevant conditional probability is defined, independence can be tested using either relationship:

or

For independent events, the multiplication rule simplifies to $P(A\cap B)=P(A)P(B)$.

In the order example, independence would imply

The observed joint probability is $0.03$, so the events are not independent in this model. The return rate for express orders is higher than the return rate overall.

Mutually exclusive events with positive probabilities are not independent. If one occurs, the other cannot, so knowing that the first occurred changes the probability of the second to zero. Independence concerns whether information about one event changes probabilities; it does not simply mean that the events are different.

Applying probability to decisions

A probability model is only as useful as its assumptions and data, so independence should be justified rather than assumed for convenience. For example, failures of two payment processors may appear separate, but a shared power outage or network problem could cause both to fail together.

If failures truly are independent and their probabilities are $0.02$ and $0.03$, the probability that both fail is

or $0.06\%$. If a shared cause creates dependence, this product understates the risk.

For a business decision, define the event clearly, identify relevant conditions, select an appropriate probability rule, and state assumptions. Probabilities can support comparisons, such as expected costs under alternative plans, but they do not by themselves determine the best decision; consequences and objectives also matter.