let's talk about a special case of two mutually exclusive events okay so we're going to focus on uh an arbitrary event a the complements of an invent a is the event of that contains all the outcomes not in a it's the event where a does not happen we write this as a with a little C like an exponent that's a complement okay so just a brief example to understand what's going on here say we're rolling a die the sample space is one two three four five six okay if you're rolling a standard six-sided die that's what you can roll that's the sample space those are the possible outcomes let's let um event m be the event of numbers bigger than four we roll a number bigger than four so that means we could either roll a five or a six then M complement is the set of all outcomes that are not in event m so what's not an aventum well that'd be one two three and four okay graphically what's going on here is if we have an event a here this is event a then everything outside of that everything that's not in a is a complement so a complement lives over here okay that's a complement so notice that together they fill the whole sample space so what do what are some properties of these complements what we've already talked about a couple first of all first of all they're mutually exclusive a and a complement two complementary events are mutually exclusive but there's more uh they fill the whole sample space right every possible outcome is an a or a complement okay so what does that mean let's look at the addition Rule and see what that's telling us okay that means that the probability that a or a complement happens because the events are mutually exclusive we could use the simplified addition rule that is we could just add the individual the probabilities of the individual events okay but we could do more we have the have the piece of information they're not just mutually exclusive they also fill the whole sample space every possible outcome is in one of these two events which means the probability that one of them happens is 100 chance there's a 100 chance that one of these two things happened if we go back to our dice example if M doesn't happen if we don't roll a 5 or 6 then we roll a 1 2 3 or 400 dice if we don't run a little one two three or four then we're rolling a five or six there's a 100 chance of one of those two things happening that's always true for complementary events so instead of just leaving this we actually know what this probability is this is 100 or 1. and so for two complementary events we could say that the addition of their probabilities will always equal one this is going to be especially useful if we're looking at something like winning and losing okay for a lot of games winning and losing are the only two possibilities okay if that's the case in their complementary events if you could only win or lose in a game then the prob I'm sorry then the event that you win is the the complement of winning is losing okay if you do not win you lose and if you do not lose you win compliment of losing is winning the complement of winning is losing again we're assuming there's only two possibilities here you win or you lose okay there are games of course where there is another option for example if you play Tic-Tac-Toe it's possible that you don't win and you don't lose it could be a draw or if you play Rock Paper Scissors maybe you both pick paper it's a draw neither win or lose that's not what we're assuming here we're assuming that you either win or lose there is no third option okay that means these two events would be complementary and therefore mutually exclusive and therefore using our previous formula if we take the probability that we win plus the probability that we lose because those are the only two options and one of those two things has happened there's a 100 chance of one of those two things happening