I've been asked about the question below on the use of conditional probability. Probability questions have more often that not being a source of pain, as I tend to find many problem statements ambiguous or confusing.
Case in point, the example below. Let's say $P(C)=0.4$ is the prob. of any household having a child, $P(M)=0.2$ that of a given household having a (>=1) male child and $P(F)=0.25$ that of having a (>=1) female child. The condition we should assume is that C is true, i.e., the households watching the show have 1 or more children. They seem to be asking then for $P(M|C)$ and $P(M\cup F|C)$.
The first part seems straightforward, once writing the above down and realizing that $M\cap C=M$:
$$P(M|C)=\frac{P(M\cap C)}{P(C)}=\frac{P(M)}{P(C)}$$
However, I don't see any information about about $M\cap F$. How then can we answer $P(M\cup F|C)$ $\big(=P(M|C)+P(F|C)-P(M\cap F|C)\big)$?
Problem statement:
You are creating a television commercial for a product which will air during a particular Saturday morning cartoon program. You have to decide whether you should market your product towards boys, girls, or both. A survey indicates that 40% of households in the area have at least one child, 20% have at least one male child, and 25% have at least one female child. If you assume that every household watching the program has a child, what is the probability that a random household which sees the commercial has a male child? A female child? Both?
EDIT: I think I now have a better grasp of the problem statement.
On one hand, I would say now I had a lack of proper understanding, as $M\cup F=C$ seems like obvious.
I have provided a possible detailed answer to this problem. Is there a shorter, yet still complete answer? What would be a key intuition leading to it?
After all these years I still struggle with written probability statements like this. I can't help but think they are all highly contrived.
In hindsight, my struggle with the wording of the problem is my own, although the ambiguity in the statement may nevertheless be seen as a legit point one could raise. We just would have to address it when providing an answer.
As mentioned we see two possibilities for interpreting the data given on the probabilities of households with male/female children: A) Those are probabilities over all households, or B) those are probabilities over all households with children. In addition, given the intrinsic ambiguity of natural language, one can interpret the last question as referring to two possible, distinct events: E1) $M\cap F$, or E2) $M\cup F$.
Hence, they ask for $P(M|C),\,P(F|C)$ and either of $P(E1|C)$ or $P(E2|C)$. The latter would trivially be 1 as $M\cup F=C$.
Note that the household with male children is a subset of all those with children, hence $M\cap C=C$.
Finally, we will need the general relation for adding the probabilities of two sets $M$ and $F$, namely $$P(M)+P(F)=P(M\cup F)+P(M\cap F)$$ which holds as well when we condition all probabilities on another event $C$.
Case A: They state that $P(C)=0.4,\,P(M)=0.2,\,P(F)=0.25$, and then $P(M|C)=\frac{P(M\cap C)}{P(C)}=\frac{P(M)}{P(C)}=1/2$ $P(F|C)=\frac{P(F\cap C)}{P(C)}=\frac{P(F)}{P(C)}=5/8$
From the relation for addition of probabilities for $M$ and $F$, $$ P(M\cap F|C)= P(M|C)+P(F|C)-P(M\cup F|C)=1/2+5/8-1=9/8-1=1/8 $$
Case B: They state that $P(C)=0.4,\,P(M|C)=0.2,\,P(F|C)=0.25$, and the problem statement is a bit silly as it provides already the answers for the first two questions. We could argue that it is a trivial exercise intended for you to just identify the information provided...or could we?
It remains, however, to calculate the probability of E1: $$ P(M\cap F|C)= P(M|C)+P(F|C)-P(M\cup F|C)=0.2+0.25-1=\,-0.45\,<\,0 $$ But probabilities are defined as a positive number! The fact that we get an absurd result means that the data does not support the interpretation of case B.
Notice here that we would not have seen this as an invalid interpretation of the problem statement had we not considered the ambiguity of the last question and calculated it as we did. We would have just concluded that the interpretation in case B renders the problem a trivial, yet possible one! A totally erroneous conclusion!!.
As mentioned, after all these years I still struggle with written probability statements like this. Maybe part of it is my reading comprehension skills, especially in a language that's not my mother tongue. But I can't help but thinking they are all highly contrived: compare this problem statement to those guided problems I would regularly find in some french (also not my mother tongue) textbooks, where they would ask you several questions and finally ask you to conclude a given claim from the previous answers. The latter follows a socratic approach guiding you to a better understanding of things, while the present one seems more the result of a lazy gamesmanship seeking to see who counts as "smarter".