the general addition rule is only used when the events are not mutually exclusive
To understand the general addition rule for probability, it's important to first understand what mutually exclusive events are. Mutually exclusive events are events that cannot occur at the same time. For example, if you toss a coin, getting a head and getting a tail are mutually exclusive events because you cannot get both heads and tails in a single coin toss.
Now, let's consider two events, A and B, that are not mutually exclusive. This means that both events can occur at the same time. For example, rolling a standard six-sided die, event A can be rolling an odd number (1, 3, or 5), and event B can be rolling a number less than or equal to 4 (1, 2, 3, or 4). In this case, events A and B can both occur if you roll a 3.
The general addition rule is used to find the probability of either event A or event B occurring or both events occurring.
The general addition rule states that:
P(A or B) = P(A) + P(B) - P(A and B)
Here's how to apply the general addition rule:
1. Determine the probability of event A occurring (P(A)).
2. Determine the probability of event B occurring (P(B)).
3. Determine the probability of both events A and B occurring (P(A and B)).
4. Apply the formula: P(A or B) = P(A) + P(B) - P(A and B).
The general addition rule is used when the events are not mutually exclusive because if they were mutually exclusive, the probability of them both occurring (P(A and B)) would be zero, leading to a simpler equation: P(A or B) = P(A) + P(B).
I hope this explanation helps! Let me know if you have any further questions.