The AND/OR Rules
The AND/OR Rules

The AND and OR rules in probability are fundamental aspects that enable us to calculate chances of different events occurring simultaneously or separately.

The OR rule is applied when we are interested in the probability of either event A or event B, or both, happening. If A and B are mutually exclusive (cannot occur together), the OR rule states that P(A or B) = P(A) + P(B).

If A and B are not mutually exclusive (can occur together), the OR rule adjusts to include the intersection of the two events happening together: P(A or B) = P(A) + P(B)  P(A and B).

The AND rule, on the other hand, is utilised when we are interested in the probability of both event A and event B happening at the same time. Here, P(A and B) = P(A) × P(B A), where P(A) is the probability of A happening and P(B A) is the conditional probability of B happening given that A has happened. 
If A and B are independent (the occurrence of A does not influence the likelihood of B), the AND rule simplifies to P(A and B) = P(A) × P(B).

In basic terms, the AND rule multiplies probabilities together whereas the OR rule adds probabilities up (with some adjustments for overlapping cases).

Lastly, remember to ensure that your calculated probabilities are reasonable. They should range from 0 (for an impossible event) to 1 (for a certain event). Anything less than 0 or more than 1 indicates an error in the process.
 Practice scenarios with these rules to improve your comfort with quick and accurate application. The ability to distinguish between situations that require the AND rule, the OR rule, or both, is paramount to achieving proficiency in this area.