A proof question involving union and complements of events

• laura_a
In summary, the equation P(X1 U X2 U X3 U ... U Xk) = 1 - [1 - P(X1)][1-P(X2)]...[1-P(Xk)] can be proven using DeMorgan's Law, which states that the complement of a union of events is equal to the intersection of the complements of each event. By applying this law, the equation can be rewritten as P(X1^c ∩ X2^c ∩ X3^c ∩ ... ∩ Xk^c), which can then be expanded using the complement rule and simplified to 1 - [1 - P(X1)][1-P(X2)]...[1-P(Xk)].

Homework Statement

If X1, X2, X3, ... Xk are independent events, prove that

P(X1 U X2 U X3 U ... U Xk) = 1 - [1 - P(X1)][1-P(X2)]...[1-P(Xk)]

The Attempt at a Solution

Well I have tried a few methods, but I know it's got something to do with

P(X1) = cc(P(X1)) (complment complement)
P(X1) = 1 - c(P(X1))
P(X1) = 1 - (1-P(X1))

I know that much, but I can't link it in with unions of independent events?? I

DeMorgan's Law:

$(A_1 \cup \dots \cup A_k)^c = A_1^c \cap \dots \cap A_k^c$

1. What is the definition of a union of events?

The union of two events, A and B, is defined as the event that either A or B or both occur.

2. How is the union of events expressed mathematically?

The union of events can be expressed as A ∪ B, where ∪ represents the union symbol. This can also be extended to more than two events, such as A ∪ B ∪ C.

3. What is the definition of a complement of an event?

The complement of an event, A, is the event that A does not occur. In other words, it is the set of outcomes that are not in A.

4. How is the complement of an event expressed mathematically?

The complement of an event can be expressed as A', where the apostrophe denotes the complement symbol. It can also be written as Ac or A¯.

5. How are unions and complements of events used in probability calculations?

Unions and complements of events are important in calculating probabilities. For example, the probability of A or B occurring can be calculated by adding the probabilities of A and B and subtracting the probability of their intersection (A ∩ B). The complement of an event can also be used to calculate the probability of the event not occurring (1 - P(A)).