关于由P(A)+P(A^c|B)=1推导事件A与B独立性的技术问询

Hey there! Let's break this down clearly to bridge the gap between your current conclusion (that $A^c$ and $B$ are independent) and the final result we need (that $A$ and $B$ are independent).

First off, your initial reasoning is totally spot-on:

• Starting with the given equation $P(A) + P(A^c|B) = 1$
• We know from basic probability axioms that $P(A) + P(A^c) = 1$
• Setting these equal tells us $P(A^c|B) = P(A^c)$, which is exactly the definition of $A^c$ and $B$ being independent (the conditional probability of $A^c$ given $B$ equals its marginal probability).

Now, to get from $A^c$ and $B$ being independent to $A$ and $B$ being independent, we can use either of two straightforward approaches:

Approach 1: Using the core definition of independent events #

Recall that two events $X$ and $Y$ are independent if and only if $P(X \cap Y) = P(X)P(Y)$. We already know $P(A^c \cap B) = P(A^c)P(B)$ from your earlier conclusion.

Note that event $B$ can be split into two mutually exclusive, exhaustive parts: $A \cap B$ and $A^c \cap B$. So:
$$P(B) = P(A \cap B) + P(A^c \cap B)$$

Rearranging to solve for $P(A \cap B)$:
$$P(A \cap B) = P(B) - P(A^c \cap B)$$

Substitute the independent condition for $A^c$ and $B$:
$$P(A \cap B) = P(B) - P(A^c)P(B) = P(B)\left(1 - P(A^c)\right)$$

Since $1 - P(A^c) = P(A)$ (complement rule), this simplifies to:
$$P(A \cap B) = P(A)P(B)$$
Which is exactly the definition of $A$ and $B$ being independent.

Approach 2: Using conditional probability directly #

We can also work with conditional probabilities. For any event $B$ with $P(B) > 0$, we know that:
$$P(A|B) + P(A^c|B) = 1$$
This holds because given $B$, either $A$ occurs or its complement does—there's no third possibility.

From the given equation, we have $P(A^c|B) = 1 - P(A)$. Substitute that into the above:
$$P(A|B) + (1 - P(A)) = 1$$
Simplify this, and we get:
$$P(A|B) = P(A)$$
This is another equivalent definition of independence: the probability of $A$ doesn't change when we condition on $B$.

Either way, we arrive at the conclusion that $A$ and $B$ must be independent.

备注：内容来源于stack exchange，提问作者goykanpravi