Calculating the Probability of Union of Independent Events A, B, and C

In probability theory, the concept of independent events plays a crucial role in calculating various probabilities. If events A, B, and C are independent and their individual probabilities are equal, then the probability of the union of these events can be determined using specific formulas. Let's explore how to calculate the probability of the union of three independent events A, B, and C.

Understanding Independent Events

Three events A, B, and C are considered independent if the occurrence of one event does not affect the probability of the others. This is a fundamental concept in probability theory. For independent events, the probability of their intersection can be calculated by multiplying their individual probabilities:

PA ∩ B PA · PB

PA ∩ C PA · PC

And,

PB ∩ C PB · PC

Intersection of the Events

Given that the probabilities of events A, B, and C are equal, i.e., PA PB PC P, we can use these to calculate their intersections:

PA ∩ B PA · PB P · P P^2

PA ∩ C PA · PC P · P P^2

PB ∩ C PB · PC P · P P^2

And,

PA ∩ B ∩ C PA · PB · PC P · P · P P^3

Using the Union Formula for Independent Events

The formula for the probability of the union of independent events A, B, and C is:

PA ∪ B ∪ C PA · PB · PC - PA ∩ B - PB ∩ C - PC ∩ A PA ∩ B ∩ C

Substituting the given probabilities and intersections:

PA ∪ B ∪ C P · P · P - P^2 - P^2 - P^2 P^3

Now, simplifying the expression:

PA ∪ B ∪ C P^3 - 3P^2 P^3

Combine like terms:

PA ∪ B ∪ C 3P - 3P^2 P^3

Conclusion

In conclusion, if A, B, and C are independent events and PA PB PC P, then the probability of the union of these events, PA ∪ B ∪ C, can be expressed as: