Does 𝑃(𝐴∪𝐵∪𝐶) always equal 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶)?
No, 𝑃(𝐴∪𝐵∪𝐶) does not always equal 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶).
The formula 𝑃(𝐴∪𝐵∪𝐶) represents the probability of the event that at least one of the events A, B, or C occurs. It is the probability that at least one of the events A, B, or C occurs, not the sum of the individual probabilities.
To calculate 𝑃(𝐴∪𝐵∪𝐶), you need to use the principle of inclusion-exclusion, which takes into account the overlapping probabilities. The formula is:
𝑃(𝐴∪𝐵∪𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) - 𝑃(𝐴∩𝐵) - 𝑃(𝐴∩𝐶) - 𝑃(𝐵∩𝐶) + 𝑃(𝐴∩𝐵∩𝐶)
Here, 𝑃(𝐴∩𝐵), 𝑃(𝐴∩𝐶), and 𝑃(𝐵∩𝐶) represent the probabilities of the intersections between the events A, B, and C.
Only when the events A, B, and C are mutually exclusive (i.e., they have no common outcomes) does 𝑃(𝐴∪𝐵∪𝐶) equal 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶). In this case, the intersection probabilities 𝑃(𝐴∩𝐵) = 𝑃(𝐴∩𝐶) = 𝑃(𝐵∩𝐶) = 𝑃(𝐴∩𝐵∩𝐶) would be zero.
No, 𝑃(𝐴∪𝐵∪𝐶) does not always equal 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶). The equation 𝑃(𝐴∪𝐵∪𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) only holds true when 𝐴, 𝐵, and 𝐶 are mutually exclusive events.
Mutually exclusive events are events that cannot occur at the same time. In such cases, the probability of the union of mutually exclusive events is equal to the sum of their individual probabilities.
For example, if 𝐴, 𝐵, and 𝐶 are mutually exclusive events, then 𝑃(𝐴∪𝐵∪𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶).
However, if 𝐴, 𝐵, and 𝐶 are not mutually exclusive events, meaning they can occur at the same time, then 𝑃(𝐴∪𝐵∪𝐶) ≠ 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶). In this case, the equation 𝑃(𝐴∪𝐵∪𝐶) = 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) needs to be adjusted using the principle of inclusion-exclusion.
Therefore, whether 𝑃(𝐴∪𝐵∪𝐶) equals 𝑃(𝐴) + 𝑃(𝐵) + 𝑃(𝐶) depends on the relationship between the events 𝐴, 𝐵, and 𝐶.