Coin tossing - what's more probable? I am solving the following probability exercise. The solution I have found is very counter intuitive and I feel It is wrong, but I can't seem to understand why.
A fair coin is tossed twice, you have to decide wheter it is more likely that two heads
showed up given that: 1) at least one toss is head, 2) the second toss was head.
Solution
Let $A$ be the event "the first toss is head" and let $B$ be
the event "the second toss is head".
For case 1,:
$$ P(A \cap B \vert A \cup B) = \frac{P(A \cap B \cap (A \cup B))}{P(A \cup B)} = \frac{P(A\cap B)}{P(A) + P(B) - P(A \cap B)} = \frac{1/4}{3/4} = 1/3$$
For case 2:
$$ P(A \cap B \vert B) = \frac{P(A\cap B)}{P(B)} = \frac{1/4}{1/2} = \frac{1}{2}$$
Is this right? I feel like case $1$ should be more probable, given that at least may mean there are already two heads?
Can someone shed some light?
 A: It is correct.
There are four equally likely outcomes (HH, HT, TH, TT) of which one outcome has two heads (HH).
In question (1) there are two other possibilities (HT, TH).
In question (2) there is only one other possibility (TH).  This makes HH conditionally more likely by excluding consideration of one of the partial successes that question (1) would consider.
A: You are correct.
If you want, have a read about the boy-girl paradox here: https://en.wikipedia.org/wiki/Boy_or_Girl_paradox
This is a very similar problem, where I tell you something along the lines of "Mr Smith has two children, and one of them is a boy". You then have to try and decode whether I intended to tell you:

*

*Either something like: I met Mr Smith's oldest child yesterday, who's a boy. This means I know the sample space of Mr Smith's children is {BG, BB} with equal probability.

*Or something like: Mr Smith was invited to an event that parents of boys only get invited to. So I know one of his children is a boy, and possibly both. So I know the sample space is {BG,BB,GB}, but with equal probability as I have no reason to suppose one combination over the other.

In the former the chance of two boys is a half, and in the later, a third.
Of course, the real problem is that natural language doesn't map nicely to the precise statements made by mathematicians (which must be partly why I never know what anyone is talking about). This is especially true about probability. When someone casually says "one of the children is a boy", they don't mean "there exists a child such that is a child of Mr Smith, and that child is a boy". But when a maths exam paper says that, it certainly is what they mean!
So in your example, statement 1) "at least one is a head", you are being told a very precise piece of information, which is weaker than saying "I saw the first toss and it was a head". You are really being told something like: "I saw a million tosses of two coins. I threw away all the examples of two tails. I randomly picked one of the remaining examples, and I am giving that example to you".
I hope this helps!
