A level conditional probability

Question

In an A Level maths textbook, the answer to an exercise question involves the use of $$P(B|A')=1-P(B|A)$$

where A and B are two events.

However, this equality has never been mentioned elsewhere in the curriculum and does not seem to yield the correct answers for conditional probability questions.

So does such an identity actually hold true or could it be just a typo?

If it is true then how can one make sense out of it?

Answer 1

It is likely that the equation that concerns you is a mistake; the closest identity (in the suitable regular sense) that holds true would be $$ P(B' \mid A) = 1 - P(B \mid A), $$ since $$ 1 - P(B \mid A) = \frac{P(A) - P(B \cap A)}{P(A)} = \frac{P(B' \cap A)}{P(A)} = P(B' \mid A). $$

The quoted equality in your book does not always hold; for instance, take $B := \varnothing$ with any $A$ such that $P(A) > 0$.

Answer 2

As noted above, they probably meant $P(B'|A) = 1 - P(B | A)$, as this actually makes sense.

It is easy to think of an intuitive counterexample. $$ P(\text{getting hit by lightning } | \text{ no thunderstorm}) = 1 - P(\text{getting hit by lightning } | \text{ thunderstorm}) $$ That would be an absurd and terrible situation.

Answer 3

The equality as written is not correct.

If e.g. $A,B$ are independent then also $A'$ and $B$ are independent so that the statement would lead to:$$P(B)=P(B\mid A')=1-P(B\mid A)=1-P(B)$$which is of course not true in general.

What they most probably meant was:$$P(B'\mid A)=1-P(B\mid A)$$which is true whenever the conditional probabilities are properly defined (i.e. if $P(A)>0$).