so we've learned now about sample space and events and outcomes and about basic probability now we're going to expand our ideas on that a little bit we're going to learn how to take two different events and to combine them to create a new event so let's start okay let a and b 2 events okay any two events in the same sample space then A and B and I'm using an uppercase and here to emphasize that we're talking about a operator between these two events and not just the word okay A and B is the event with all the outcomes that are in both A and B at the same time okay so let's see a quick example here let's say a is the event with outcomes one two three four okay and B is an event and the outcomes in event B are uh 2 4 6 and 8. okay then the vent A and B okay and and we should think about this as a unit it is a new event created with these two but it is just one event that we've used that we've created by combining these two events okay event A and B are all outcomes that are in both of the original events okay so which outcomes are in both of the events well um if we look at one one is an event a but it's not an event B so one would not be included in a and b because it's not in both of them okay but two is in both events okay so 2 would be in the event A and B okay three it's not in both events it's an event a but not b but four is in both events because it's in both events it's in a and b and that the those are the only two outcomes that are in both event a and in event B okay and so the new event A and B is the event with outcomes two and four okay we can visualize this as a kind of a Venn diagram okay so let this be our entire sample space okay this represents a Vente all the outcomes are nervente are in this circle okay and this represents event B all the outcomes in event B are in this circle A and B the new event A and B is represented by this portion in the middle here color this portion here so in a Venn diagram sense it's everything that they have in common okay so this right here that is event a and B okay okay so that's uh the and operator there's one more operator that we need to consider that's the or operator so we're going to start off with the definition let a and B 2 events in the same sample space then the event A or B and again notice I'm capitalizing this just to emphasize that this is more than just a word it is a mathematical operator that's taking these two original events and combining them in a new way okay then the event A or B is the event with all the outcomes that are either in a or in b or both it's okay to be in both okay so let's take a look at the same two events we were looking at earlier so say event a has outcomes one two three four just like before and event B has outcomes 2 4 6 8. okay then what would a or b look like what would this new event and again this is just one event that we get by combining these two using the or operator this new event what's in it okay what we want all the outcomes that are either an A or in b so we want to include one and two and three and four because they're an A so everything in a gets included one two three four and everything in b gets included 2 4 6 8. now two of these I've already included right two and four they're already here I don't need to include them again each outcome only gets counted once even if it's in both the events that we're combining okay so I'm not going to count two and four again they've already been counted they've already been placed into the new event but I do want to make sure I include six and eight Okay so now every outcome that is in here or that is in here is in the new event A or B okay if we want to consider this like a Venn diagram again let this rectangle be the sample space okay and we'll have two circles here this circle represents event a this circle represents event B then the new event A or B is all of the outcomes in a and all of the outcomes in b okay it's all of this stuff okay so both the circles when you put them together that gives you a or B those two circles together are the combined event A or B