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We've looked at how to find the probability of a single event occurring, but now we want to start talking

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about the probability of multiple events, and we're going to start that investigation with this idea

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of the addition rule.

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So the addition rule, technically, if we look at its formula, is this rule here.

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So we say the probability of A or B, So this symbol right here, it's like a U-shaped symbol we can

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interpret this to mean or so this probability here is the probability that A or B occurs.

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Contrast that with this upside down U right here, which we interpret to mean.

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And so this probability is the probability of A and B both occurring.

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This symbol, the U.

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We call union and this symbol we call intersection.

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So this is union, this is intersection union we can think of as or intersection we can think of as.

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And so another way to write this formula is this way where we replace the union symbol with or and we

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replace the intersection symbol with and otherwise these formulas are identical.

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We've just written the union and intersection symbols differently here.

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We've used these symbols here we've used the words or an and but the formulas mean exactly the same

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thing.

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So either way, as we're looking through this, what we're saying is the probability that A or B occurs

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is equivalent to the probability of a occurring plus the probability of B occurring minus this intersectional

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probability, the probability that both A and B occur at the same time.

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Now, approaching this directly from the formula is a little confusing, but if we approach it from

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a real world scenario, it actually makes a lot of sense.

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So let's do that now.

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And in order to look at this from a real world perspective, we need to think about two different types

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of events A and B, we can have the situation where the events A and B are what we call mutually exclusive

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or disjoint, or we can have the scenario where the events A and B are overlapping.

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For example, let's think about the mutually exclusive events of rolling a single six sided die.

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And considering the probability that we get one or an even number when we roll that die, those are

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mutually exclusive events.

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Because if we think about it this way, if we have a single six sided die and we roll it one time and

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we say that event A is getting A one, an event B is getting an even number.

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Well, the only die roll that results in one is this roll right here.

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Rolling a one is the only outcome that matches event a of getting a one when we roll the die.

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But if event B is getting an even roll on the die, that could happen when we roll a two, when we roll

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a four or when we roll a six so we can satisfy event A by rolling a one, we can satisfy event B by

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rolling a two of four or a six.

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What we want to realize here then is that events A and events B are mutually exclusive because they

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have no overlap between them.

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There's no overlap between these sets one and then separately two, four and six.

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If we had one occurring in both sets or two occurring in both sets or four or six, if we had any commonality

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between these two sets, then events A and B would be overlapping.

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But here there's no overlap between these two sets and so events A and B are mutually exclusive.

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And so this example where if we roll a six sided die one time and we're looking for the probability

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that we get either one or an even number, those two events, their events A and B are mutually exclusive.

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They don't overlap.

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And so one way we can visualize that is with this Venn diagram, you may already be somewhat familiar,

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at least with this idea of a Venn diagram where we display the outcomes for any particular event within

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one circle in the Venn diagram.

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So for this particular example, if we think about the circle on the left as representing event A,

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which is the event that we roll a one and this circle on the right as representing event B where we

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roll.

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An even number, then what we can see here is that there is one outcome that matches our criteria for

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event A, So we would put a one in this event a circle.

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There are three outcomes that match our criteria for event B, and so we would put a three in this event

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B circle.

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We know that there are six possible outcomes when we roll a six sided die one time.

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And so what's probably pretty intuitive to us is that the probability that we roll either a one or an

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even number is going to be one plus three divided by six or.

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For over six or.

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Two over three.

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So we have a two thirds chance, a two in three chance, or an approximately 67% chance of rolling a

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one or an even number.

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If we roll a six sided die one time, that probably feels pretty straightforward.

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The way that this actually applies to our addition rule formula that we started with is this way here.

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So we'll put it over here.

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On the right side, we can say that the probability of event A or event B is the probability of rolling

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a one or an even.

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Number and that that is equal to the probability that event a occurs.

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Well, we know that that is one over six possibilities.

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So we would say one over six plus the probability of event B occurring, which we know is three over

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six, so three over six.

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And then we would subtract what we call the intersection, the intersection probability of both of these

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events occurring.

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But when we have mutually exclusive events, that intersection of probability is zero.

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And we see that in our Venn diagram because these circles don't overlap in this little chart that we

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built, there's no overlap between the set one and the set two, four and six, so there's no overlap

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in our chart.

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Our circles don't overlap.

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There's no scenario where we can satisfy rolling a one and rolling an even number at the same time.

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There's zero probability that event A and event B occur at the same time.

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It's impossible to roll both a one and an even number when we roll a six sided die one time.

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And so this probability is zero.

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So we would just subtract zero and the result then would be one over six plus three over six is four

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over six or two thirds, just like we got down here when we did this intuitively.

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In other words, when we have mutually exclusive events, when the events are disjoint, when they have

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no overlap, when it's impossible for both events to occur at the same time, then this probability

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here where this probability here is zero.

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And so our addition rule formula simplifies to just the probability of A or B is equal to the probability

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of A plus, the probability of B, And we completely ignore this term because it just goes to zero.

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And so we're subtracting zero.

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It has no effect on the formula.

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And so our addition rule formula just becomes this formula here and nothing else.

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That's when we have mutually exclusive events, but when we have overlapping events.

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So let's change the scenario here to the scenario where we roll a six sided die one time and we're looking

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for the probability that we get a one or an odd number.

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So event A is rolling, A one event B is rolling an odd number.

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If we build out our little chart here, just like we did before, what we can say is that event a rolling

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a one, there's only one possible outcome there that we're interested in and it's this outcome rolling

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a one this is here event A and this is event B OC but then rolling an odd number will roll an odd number

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if we roll either a one, a three or a five, because those are the odd numbers on our six sided die.

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And what we see right away is that we have an overlap for the first time.

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We have the overlap of one in both sets here.

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So set A is just one, set B is one, three and five.

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So we have this overlapping value of one in both sets A and B, and that makes sense because one is

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an odd number.

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So we can satisfy event A by rolling a one we can satisfy event B by rolling a13 or five and one occurs

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in both sets, which means that this intersectional probability, the probability of A and B both occurring

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is non zero.

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Our events are overlapping.

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There is some probability, of course, that we could roll a one and an odd number because one itself

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is an odd number.

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When we have overlapping probability like this, when we have overlap in our table, when our intersectional

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probability is non zero, then in our Venn diagram we have overlapping circles and this overlapping

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portion in the center here, the overlapping part of the circles represents that overlap.

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So if we label this left circle again as event, a occurring being a one and this right side circle

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as event B being an odd number, then we need to treat each of the three sections of our Venn diagram

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very carefully.

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So this section on the left here, the section that is inside circle A here, but completely outside

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of circle B, So just this little moon shaped section over here on the left.

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This section represents all of the possibilities where we roll a one, but not an odd number.

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Think back to when we talked about complementary probability.

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This right side circle represents all of the odd numbers.

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So when we're outside of this right side circle, we're completely outside of the odd circle and we're

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just in this little left hand portion here.

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These are all the outcomes where event occurs, but event B definitely does not occur.

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So this left side here is A not B, this right side is.

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Not a and the middle is A and B.

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So if we're thinking about the left side as a not B, then we're saying all of the outcomes where we

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roll a one, but not an odd number.

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But of course that's zero because it's impossible to roll a one and not an odd number one is itself

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an odd number.

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So if we were to roll a one, by definition we also roll an odd number.

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So this has to be zero because there's literally no way to roll a one and not an odd number.

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So this section over here of A not B contains zero possibilities.

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On the right hand side here, this is B, not A, So the probability that we roll an odd number, but

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not a one is two.

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Because if we go back to our chart here, there are three ways we can roll an odd number one, three

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and five.

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But if we want an odd number and not a one, we can't include this one value in our set one, three

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and five.

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So the only way to satisfy B and not A or an odd number and not one is with a three or a five.

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So there's two possibilities.

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So we put two over here and then in the center this is A and B both occurring.

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So how many ways are there to get a one and an odd number?

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Well, there's only one way because there's only one way to get a one, but one is itself an odd number.

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So there's exactly one and only one way that we can roll a one and an odd number, and that is to roll

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a one itself.

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So we fill in our Venn diagram and then if we want to use this Venn diagram to determine the probability

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that A or B occurs, going back to our addition rule formula, we want the probability of A or B, the

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probability of A or B, all we have to do is follow the formula and look at the probability of a well,

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the probability of a If we look at the entire circle of a here, there's one possibility that a occurs.

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In other words, we add up all of the values that are inside the circle A so zero plus one or one.

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There's one possibility that A occurs.

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We know there's six total possibilities when we roll a six sided die one time.

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So the probability of a occurring is one over six, the probability of event B occurring.

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We add up all the values inside the circle.

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B that's one plus two or three.

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So we say plus three over six, but then we subtract this intersectional probability.

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That means subtracting the intersection of the Venn diagram.

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And inside the intersection here we have this value of one.

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So we subtract one over six and we get one over six plus three over six minus one over six or one plus

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three minus one all over six is equal to.

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Three over six or one half.

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So the probability of A or B occurring in this scenario is one half.

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The probability of rolling a one or an odd number is one half.

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And that makes sense because when we look back at our chart here, the probability that we get a one

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or an odd number just intuitively is going to be satisfied when we roll a one or a three or a five,

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which is exactly half of the possibilities when we roll the six sided die one time.

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So specifically plugging into this addition rule formula here, the probability of getting a one or.

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An odd number is one over six plus.

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Three over six minus.

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And then instead of our intersectional probability being zero like it always is with mutually exclusive

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events, when we have overlapping events, we subtract the intersection, which in our case is one over

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six.

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So we get minus one over six is equal to one plus three minus one over six or three over six.

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Three over six is equal to one half.

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And so the probability of getting a one or an odd number is one half the other way.

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We can calculate this probability when we have overlapping events.

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It's just to add up all the numbers in our Venn diagram so we can just add zero plus one plus two to

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get three.

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Three is our number of matching events.

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We divide that by six, the number of total events, and so we get three over six or one half the same

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value we got here.

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In other words, two ways to calculate the probability here.

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If we've built out the Venn diagram, we can just add all the numbers in the Venn diagram.

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Zero plus one plus two is three three divided by six.

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We get our probability of one half or we can add up all the values in Circle A to get the probability

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of a add up all the values in circle B to get the probability of B, and then subtract out the intersectional

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probability to eliminate that overlap.

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And we'll get to the same answer three over six or one half, 50% chance of rolling a one or an odd

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number.

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So again, the whole idea here with this addition rule and this union versus intersection concept is

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just that we have to pay special attention to or be careful with this intersectional probability right

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here, because when we're calculating the probability of one event or another occurring, we don't want

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to double count events that occur in both sets over here with this overlapping probability, one occurred

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in both set A and set B, And so if we don't subtract out this intersectional probability, if we don't

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subtract out this value, we will double count it as part of event A and part of event B and will overstate

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our probability.

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Going back to our chart here, if we just say that the probability of event A is one over six.

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So this event right here is one over six and the probability of event B is three over 6123 matching

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outcomes divided by six possible outcomes.

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So plus three over six.

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And we just add those two probabilities together we get four over six or two thirds or a 67% chance

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that we get one or an odd number.

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But of course we know already that that's not true.

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The probability of getting one or an odd number is one half.

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It's 50%, not two thirds or 67%.

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This calculation where we just add the two individual probabilities together, double counts this value

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of one in both probabilities.

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And so we overstate the probability.

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We have to account for the fact that one occurs in both sets.

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And we need to make sure that we don't double count it and instead that we only count it one time eliminating

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the overlap here or taking out that intersectional probability where we subtract out the probability

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of A and B occurring, we subtract out the probability of A and B occurring.

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We eliminate that double counting and we end up with an actually accurate probability for this scenario

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of overlapping events.

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So that's how we use the addition rule to calculate the probability that event A or event B occurs by

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using this concept of union and intersection and translating that into this visual idea of the Venn

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diagram where we can clearly show mutually exclusive events with non overlapping circles or overlapping

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events with these overlapping circles in our Venn diagram.