What is the intuition behind $P(A\cap B) = P(A)P(B)$, given A and B are independent events?   I saw the derivation but wasn't able to to think of it intuitively. Is there any combinatorial reasoning behind it?  I have even seen it for cases like tossing a coin and it works! But I am not convinced that it should be true always. I am looking for a strong foundation of it.
I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and I see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$.    But the first statement is not normally taken as a definition; instead the second is.    What is the intuition, or even derivation behind defining $A$ and $B$ as independent iff $P(A, B) = P(A)(B)$?    The kind of explanation I am looking for would be one similar to that given by Jaynes for the definition of conditional probability in the first chapter of Probability: The Logic of Science, or even a Kolmogorov axiomatic explanation would help.
When you're logged in to a Stack Exchange account, it's possible to add and remove logins (associated OpenID/OAuth accounts) without performing any form of reauthentication.  This is not a full vulnerability in itself, but it provides an extremely easy way for an attacker who's found a temporary XSS vulnerability to turn that into a permanent account hijack. An XSS exploit on its own may only provide a minute of indirect access to a user's account. But with this login configuration, the attacker could use that minute to automatically:   Add a new attacker-controlled login to the account. For example:   The XSS JavaScript would generate a random identifier, and use that to construct a login URL of the form http://hacker.example/openid/1234567. The hacker.example server would automatically authenticate the first OpenID login request made with a given identifier, allowing it to be added to the account without user interaction. All subsequent login attempts would be automatically rejected for the user. The hacker.example server would notify the operator with the account name and the generated identifier, and would allow them to login to it.  Remove all existing logins from the account. Log the user out of their account, with no way to get back in.   If this exploit were used to hijack moderator and staff accounts it could makes it more difficult to quickly respond to an attack, creating a greater opportunity for extended abuse.  This is a broader instance of the same fundamental vulnerability which was previously-reported as . That instance is a little more straightforward to fix: just ask for the password again when changing it. However, safe reauthentication with OpenID/OAuth logins is more complicated. Most users' OpenID providers will be configured to automatically re-authenticate upon request, so there would be minimal security gain from requesting that.  Instead or as well, my suggestion is that for users that have a verified email address, require an emailed confirmation link to be clicked before committing any login changes.    I'm not a real security expert, but I've just found that fellow Stack Overflow user , who is, previously described this vulnerability in his 2012 blog post :     If an attacker can add additional login accounts then they can create a persistent mechanism to get back into that Stack Overflow account at any time without needing to hijack the session. Now of course the victim can always go into their account settings and identify the presence of the new login, but it’s a question of how much damage is done before then.      How would an attacker mitigate the risk of having their newly added account removed? Just remove the victim’s original account(s) instead.      [...]      Finally, this is also a lesson for those considering using third party authentication providers. 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