# Prove that the probability of two event sets are equal

Consider this problem:

Let $A_1, A_2,...$ be an arbitrary finite sequence of events. Let $B_1, B_2,...$ be another finite sequence of events defined as follows: $B_1 = A_1, B_2 = A^c_1 >\cap A_2, B_3 = A^c_1 \cap A^c_2 \cap A_3,..$

Prove: $P(\bigcup^n_{i=1} A_i) = \sum^n_{i=1}P(B_i)$

Proof using Induction:

For $\mathbf{n=1}$ $$P(\bigcup^1_{i=1} A_i) = P(A_1) \ \text{and} \ \sum^1_{i=1}P(B_i) = P(B_1)$$ $$\text{Given} \ A_1 = B_1 \ \text{it follows that} \ P(A_1) = P(B_1)$$

Inductive step:
Assumption: $$\forall n \in N \ | \ P(\bigcup^n_{i=1} A_i) = \sum^n_{i=1}P(B_i)$$ prove that: $$P(\bigcup^{n+1}_{i=1} A_i) = \sum^{n+1}_{i=1}P(B_i)$$

Prove: $$P(\bigcup^{n+1}_{i=1} A_i)$$ $$\iff P(\bigcup^{n}_{i=1} A_i \cup A_{n+1})$$ $$\iff P(\bigcup^{n}_{i=1} A_i) + P(A_{n+1}) - P(\bigcup^{n}_{i=1} A_i \cap A_{n+1})$$

$$\iff P(\bigcup^{n}_{i=1} A_i) + P(A_{n+1} \cap (\bigcup_{i=1}^n A_1)^c)$$

Applying De Morgan Law

$$\iff P(\bigcup^{n}_{i=1} A_i) + P(A_{n+1} \cap (\bigcap_{i=1}^n A_1^c))$$

Due to preexistence of $B_n = \bigcap_{i=1}^{n-1} A_i^c \cap A_{n} \ \text{for} \ n > 1$ and therefore $B_{n+1} = \bigcap_{i=1}^n A_i^c \cap A_{n+1}$ it follows that

$$\iff P(\bigcup^{n}_{i=1} A_i) + P(B_{n+1})$$

Employing our inductive assumption for $\forall n \in N$ it follows:

$$\iff \sum^n_{i=1} P(B_i) + P(B_{n+1}) \iff \sum^{n+1}_{i=1} B_i$$

The proof at some stages does make sense in my head. Could someone please tell me whether those steps are consistent?
Thank you.

When you assume the induction hypothesis, you assume that there is some $n$ that satisfies the property $P$. You say $\forall n$, but it should be $\exists n$. Because you're ultimately proving that it's true for all $n$. Until you do that, assume it's true for some $n$.
Next, your use of $\iff$ is wrong, I assume you mean equality. You use $\iff$ to connect sentences, but you are comparing numbers, not sentences.
When you say "pre-existence of $B_n$", I'm not sure what you mean. Of course $B_n$ exists, otherwise there's nothing to prove. You probably mean "because we assumed the induction hypothesis".
If the symbols $\forall, \exists, \iff$ etc confuse you, it's okay to use natural language.