The condition in conditional probability

Question

I'm given the statement:

"A survey of Australians living outside of Melbourne or Sydney has found that 25% have not visited Melbourne, 15% have not visited Sydney, and 10% have never been to Sydney or Melbourne, A person is selected at random."

with the following questions:

a) If she has not visited Melbourne, what is the probability that she has not visited Sydney?

b) If she has not visited Sydney, what is the probability that she has not visited Melbourne?

The answers provided are as follows:

Probability of Not Visiting Melbourne = $P(M) = 0.25$

Probability of Not Visiting Sydney = $P(S) = 0.15$

$P(M \cap S)$ = 0.10

I'm wondering why we are using an intersection here, as the question says "never been to Sydney or Melbourne", and not "Sydney and Melbourne"?

Furthermore,

The answer for A has been given as: a) "The probability that a person has not visited Melbourne given that she has not visited Sydney" = P(M|S) = ....

b) "The probability that a person has not visited Sydney given that she has not visited Melbourne" = P(S|M) = ...

Now, I'm having trouble seeing how a) is the answer for a), and b) is the answer for b). In my eyes, they should be the other way around...

Here's why:

a) "If she has not visited Melbourne, what is the probability she has not visited Sydney? b) If she has not visited Sydney what is the probability she has not visited Melbourne?

In my eyes, the condition is what immediately follows the "if". Yet, for a) the condition is NOT what follows the if, and for b), the condition is NOT what follows the if.

So, I don't know if this is a mistake on their part, or if I am confused in what the condition is in Conditional Probability...

Can anyone please help me with resolving this issue?

Thanks.

Answer 1

Regarding your first question (the use of the intersection rather than the union of the events $M$ and $S$), this is an instance of one of De Morgan's laws. The event 'the selected person has never been to Sydney or Melbourne' means that it is not true that she has ever been to either of the cities Sydney or Melbourne. In other words, she has not been to Sydney, nor has she been to Melbourne. Symbolically, we write $(M^c \cup S^c)^c = M \cap S$. Here, the superscript $^c$ is used to denote the complementary of the respective event.

Regarding the second part of your question, you are right, the answers have been mixed up.

Answer 2

OK, if I understand the question correctly, 'never been to Sydney or Melbourne' is event $S \cap M$. Your first question asks for $P(S|M)$, the second $P(M|S)$. Now keep in mind two things:

1) $P(A \cap B) = P(B \cap A)$

2) $P(A \cap B) = P(B |A)P(A)$

Answer 3

"10% have never been to Sydney or Melbourne" means 10 % have neither been to Sydney nor Melbourne. Also, the other interpretation (you provided) would violate this $n(M \cup S) = n(M) + n(S) - n(M \cap S) $, with $n(M) = 0.25, n(S) = 0.15$. Obviously, $n(M \cup S)$ has to be greater than or equal to $n(M)$. Given this I think the rest of questions are substitution problems.