Suppose you want to know the probability that you draw a jack or a red card from a standard 52-card deck. You know there are four jacks in the deck, one of each of the four suits (hearts, diamonds, clubs and spades). There are 13 cards of each color, so there are 26 red cards.
So one might think the probability is 4/52 for the jacks plus 26/52 for the red cards to get 30/52. But that isn't correct because if you do that you care basically counting the jack of hearts and jack of diamonds twice, since they are in both sets of outcomes.
To fix that, you have to subtract 2 from the total to get 28/52, simplified to 7/13.
The basic formula for two events that can both occur simultaneously, you take P(First event) + P(Second event) - P(intersection of the events).