P(A or B) = P(A) + P(B) − P(A and B) when
1. A and B are mutually exclusive
2. A and B are complements
3. A and B occur together
4. A and B are dependent
I am pretty confused on this one help will be great :)
No problem, I'll be glad to help you understand this!
The formula you mentioned, P(A or B) = P(A) + P(B) - P(A and B), is known as the inclusion-exclusion principle. It helps us calculate the probability of either event A or event B occurring, taking into account their individual probabilities as well as the probability of both events A and B occurring together.
Now, let's go through each scenario to see how the formula applies:
1. A and B are mutually exclusive: In this case, it means that events A and B cannot occur at the same time. For example, if A represents "flipping a coin and getting heads" and B represents "flipping a coin and getting tails," then A and B are mutually exclusive. In this situation, P(A and B) would be 0, since the two events happening simultaneously is impossible. Therefore, the formula simplifies to P(A or B) = P(A) + P(B).
2. A and B are complements: When A and B are complements, it means that they are mutually exclusive and together they cover all possibilities. For example, let's say A represents "rolling a die and getting an even number" and B represents "rolling a die and getting an odd number." In this case, A and B are mutually exclusive, and rolling a die will always result in either an even or an odd number. Therefore, P(A and B) = 0, and the formula simplifies to P(A or B) = P(A) + P(B).
3. A and B occur together: If events A and B occur together, it means they are not mutually exclusive. For example, let's say A represents "choosing a red card from a deck of cards" and B represents "choosing a face card from a deck of cards." In this case, A and B can both happen simultaneously, as there are red face cards in a deck. In this scenario, P(A and B) represents the probability of both A and B happening at the same time, and it is calculated separately. The formula remains P(A or B) = P(A) + P(B) - P(A and B).
4. A and B are dependent: When events A and B are dependent, it means that the occurrence of one event affects the probability of the other event. In this case, P(A and B) would depend on the context and you would need additional information to calculate it accurately. The formula still applies as P(A or B) = P(A) + P(B) - P(A and B), but you would need to determine the value of P(A and B) based on the specific relationship between A and B.
I hope this explanation helps clarify how the inclusion-exclusion principle works in different scenarios. If you have any additional questions, feel free to ask!