I am new to probability, and I am finding it very difficult to understand, the probability of events A and B, and A given that B occurred. I was solving a problem and I got it completely wrong because I thought it was taking A and B but it is actually A given B. Here is the piece of the problem
Suppose that 65 percent of the Mac users have succumbed to a computer virus, 82 percent of the Windows users get the virus, and 50 percent of the Linux users get the virus.
I thought it was talking that 65% chance that user is a mac user and he is infected by a virus, but it was actually 65% chance that he was infected by a virus given that he is a mac user.
I know the formula P(A|B)=P(A and B)/P(B), But I find the above two statements completely same. That he is a mac user and he is infected by a virus, and he is infected by virus given that he is a mac user.
Can you please explain the difference? And how can I find out whether it is talking about A and B, or A|B?
Thank you for reading the question.
In general, the key parts of the sentence are the value of our probability (65%), the population it's with respect to ("of users," vs "of Mac users"), and the conditions they meet ("are Mac users who got the virus" vs "got the virus"). Conditions go in the first part of the probability, populations go after the bar symbol (or are ignored if it's "everyone").
The sentence " 65% of the Mac users get the virus " is a conditional probability - $\mathbb P (A | B) = \frac {65} {100}$ - it's saying that if we only look at the people who are Mac users, 65% will have the virus.
The sentence " 65% of computer users are Mac users who got the virus " is indicating the probability of someone being in two groups at once (both a MAC user, and getting the virus) - $\mathbb P (A \cap B) = \frac {65}{100}.$
In general, mathematics doesn't like to waste words. If you see something that's written in a convoluted way for what you think they mean, take a second to think about what else they could be trying to say. Usually (though not always), the reader has missed a definition or misunderstood a subtlety.