Probability Theory and Set Concepts

Intersection of A and B

The intersection of $A$ and $B$, denoted as $A \cap B$, is the event containing only the sample points that belong to both $A$ and $B$. This represents the "overlap" between the two events.

Mutually Exclusive Events and the Addition Law

Not all events overlap. Some events are mutually exclusive, meaning they cannot occur at the same time. For example, if you are selecting a single winner from a group of restaurant types, a winner cannot be both a "fast food" restaurant and a "buffet" simultaneously.

When events are mutually exclusive, we use the General Additional Rule (or Addition Law) to find the probability that either event $A$ or event $B$ occurs:

$P(A \cup B) = P(A) + P(B)$

Worked Example: Restaurant Winners

Imagine a business convention is selecting a random winner from a group of 23 restaurants. The group includes 5 fast food, 3 casual dining, 2 fine dining, 6 family-style, 3 food trucks, 3 cafes, and 1 buffet.

If we want to find the probability of selecting either a fast food winner ($A$) or a buffet winner ($B$):

1. $P(A) = \frac{5}{23}$
2. $P(B) = \frac{1}{23}$
3. $P(A \cup B) = \frac{5}{23} + \frac{1}{23} = \frac{6}{23} \approx 0.2609$ or $26.09%$.

Conditional Probability

Often, we want to know the likelihood of an event occurring based on the knowledge that another event has already happened. This is known as conditional probability, denoted as $P(A \mid B)$.

In conditional probability, we are no longer looking at the entire sample space; instead, we are only interested in the portion of $A$ that intersects with $B$. For this to be mathematically possible, the probability of the condition, $P(B)$, must not be zero.

The formulas for conditional probability are:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

$P(B \mid A) = \frac{P(A \cap B)}{P(A)}$

Exam Tip: When calculating $P(A \mid B)$, remember that $B$ is your new "universe." You are dividing the probability of both events happening by the probability of the condition itself.

Worked Example: Calculating Likelihoods

Suppose we have two events, $A$ and $B$, with the following probabilities:

• $P(A) = 0.40$
• $P(B) = 0.70$
• $P(A \cap B) = 0.20$

To find the probability of $A$ given $B$: $P(A \mid B) = \frac{0.20}{0.70} \approx 0.2857$

To find the probability of $B$ given $A$: $P(B \mid A) = \frac{0.20}{0.40} = 0.50$

Advanced Application: Bayes' Theorem

Building upon conditional probability, Bayes' Theorem allows managers and leaders to update their beliefs as new evidence becomes available. This involves looking at a "prior" probability (what we believed before the evidence) and using new data to reach a "posterior" probability (our updated belief).

This can be visualized using a Bayes Decision Tree or a Bayes Table to organize prior probabilities, conditional probabilities, and joint probabilities to reach a final decision.