We've looked at how to find the probability of a single event occurring, but now we want to start talking
about the probability of multiple events, and we're going to start that investigation with this idea
of the addition rule.
So the addition rule, technically, if we look at its formula, is this rule here.
So we say the probability of A or B, So this symbol right here, it's like a U-shaped symbol we can
interpret this to mean or so this probability here is the probability that A or B occurs.
Contrast that with this upside down U right here, which we interpret to mean.
And so this probability is the probability of A and B both occurring.
This symbol, the U.
We call union and this symbol we call intersection.
So this is union, this is intersection union we can think of as or intersection we can think of as.
And so another way to write this formula is this way where we replace the union symbol with or and we
replace the intersection symbol with and otherwise these formulas are identical.
We've just written the union and intersection symbols differently here.
We've used these symbols here we've used the words or an and but the formulas mean exactly the same
thing.
So either way, as we're looking through this, what we're saying is the probability that A or B occurs
is equivalent to the probability of a occurring plus the probability of B occurring minus this intersectional
probability, the probability that both A and B occur at the same time.
Now, approaching this directly from the formula is a little confusing, but if we approach it from
a real world scenario, it actually makes a lot of sense.
So let's do that now.
And in order to look at this from a real world perspective, we need to think about two different types
of events A and B, we can have the situation where the events A and B are what we call mutually exclusive
or disjoint, or we can have the scenario where the events A and B are overlapping.
For example, let's think about the mutually exclusive events of rolling a single six sided die.
And considering the probability that we get one or an even number when we roll that die, those are
mutually exclusive events.
Because if we think about it this way, if we have a single six sided die and we roll it one time and
we say that event A is getting A one, an event B is getting an even number.
Well, the only die roll that results in one is this roll right here.
Rolling a one is the only outcome that matches event a of getting a one when we roll the die.
But if event B is getting an even roll on the die, that could happen when we roll a two, when we roll
a four or when we roll a six so we can satisfy event A by rolling a one, we can satisfy event B by
rolling a two of four or a six.
What we want to realize here then is that events A and events B are mutually exclusive because they
have no overlap between them.
There's no overlap between these sets one and then separately two, four and six.
If we had one occurring in both sets or two occurring in both sets or four or six, if we had any commonality
between these two sets, then events A and B would be overlapping.
But here there's no overlap between these two sets and so events A and B are mutually exclusive.
And so this example where if we roll a six sided die one time and we're looking for the probability
that we get either one or an even number, those two events, their events A and B are mutually exclusive.
They don't overlap.
And so one way we can visualize that is with this Venn diagram, you may already be somewhat familiar,
at least with this idea of a Venn diagram where we display the outcomes for any particular event within
one circle in the Venn diagram.
So for this particular example, if we think about the circle on the left as representing event A,
which is the event that we roll a one and this circle on the right as representing event B where we
roll.
An even number, then what we can see here is that there is one outcome that matches our criteria for
event A, So we would put a one in this event a circle.
There are three outcomes that match our criteria for event B, and so we would put a three in this event
B circle.
We know that there are six possible outcomes when we roll a six sided die one time.
And so what's probably pretty intuitive to us is that the probability that we roll either a one or an
even number is going to be one plus three divided by six or.
For over six or.
Two over three.
So we have a two thirds chance, a two in three chance, or an approximately 67% chance of rolling a
one or an even number.
If we roll a six sided die one time, that probably feels pretty straightforward.
The way that this actually applies to our addition rule formula that we started with is this way here.
So we'll put it over here.
On the right side, we can say that the probability of event A or event B is the probability of rolling
a one or an even.
Number and that that is equal to the probability that event a occurs.
Well, we know that that is one over six possibilities.
So we would say one over six plus the probability of event B occurring, which we know is three over
six, so three over six.
And then we would subtract what we call the intersection, the intersection probability of both of these
events occurring.
But when we have mutually exclusive events, that intersection of probability is zero.
And we see that in our Venn diagram because these circles don't overlap in this little chart that we
built, there's no overlap between the set one and the set two, four and six, so there's no overlap
in our chart.
Our circles don't overlap.
There's no scenario where we can satisfy rolling a one and rolling an even number at the same time.
There's zero probability that event A and event B occur at the same time.
It's impossible to roll both a one and an even number when we roll a six sided die one time.
And so this probability is zero.
So we would just subtract zero and the result then would be one over six plus three over six is four
over six or two thirds, just like we got down here when we did this intuitively.
In other words, when we have mutually exclusive events, when the events are disjoint, when they have
no overlap, when it's impossible for both events to occur at the same time, then this probability
here where this probability here is zero.
And so our addition rule formula simplifies to just the probability of A or B is equal to the probability
of A plus, the probability of B, And we completely ignore this term because it just goes to zero.
And so we're subtracting zero.
It has no effect on the formula.
And so our addition rule formula just becomes this formula here and nothing else.
That's when we have mutually exclusive events, but when we have overlapping events.
So let's change the scenario here to the scenario where we roll a six sided die one time and we're looking
for the probability that we get a one or an odd number.
So event A is rolling, A one event B is rolling an odd number.
If we build out our little chart here, just like we did before, what we can say is that event a rolling
a one, there's only one possible outcome there that we're interested in and it's this outcome rolling
a one this is here event A and this is event B OC but then rolling an odd number will roll an odd number
if we roll either a one, a three or a five, because those are the odd numbers on our six sided die.
And what we see right away is that we have an overlap for the first time.
We have the overlap of one in both sets here.
So set A is just one, set B is one, three and five.
So we have this overlapping value of one in both sets A and B, and that makes sense because one is
an odd number.
So we can satisfy event A by rolling a one we can satisfy event B by rolling a13 or five and one occurs
in both sets, which means that this intersectional probability, the probability of A and B both occurring
is non zero.
Our events are overlapping.
There is some probability, of course, that we could roll a one and an odd number because one itself
is an odd number.
When we have overlapping probability like this, when we have overlap in our table, when our intersectional
probability is non zero, then in our Venn diagram we have overlapping circles and this overlapping
portion in the center here, the overlapping part of the circles represents that overlap.
So if we label this left circle again as event, a occurring being a one and this right side circle
as event B being an odd number, then we need to treat each of the three sections of our Venn diagram
very carefully.
So this section on the left here, the section that is inside circle A here, but completely outside
of circle B, So just this little moon shaped section over here on the left.
This section represents all of the possibilities where we roll a one, but not an odd number.
Think back to when we talked about complementary probability.
This right side circle represents all of the odd numbers.
So when we're outside of this right side circle, we're completely outside of the odd circle and we're
just in this little left hand portion here.
These are all the outcomes where event occurs, but event B definitely does not occur.
So this left side here is A not B, this right side is.
Not a and the middle is A and B.
So if we're thinking about the left side as a not B, then we're saying all of the outcomes where we
roll a one, but not an odd number.
But of course that's zero because it's impossible to roll a one and not an odd number one is itself
an odd number.
So if we were to roll a one, by definition we also roll an odd number.
So this has to be zero because there's literally no way to roll a one and not an odd number.
So this section over here of A not B contains zero possibilities.
On the right hand side here, this is B, not A, So the probability that we roll an odd number, but
not a one is two.
Because if we go back to our chart here, there are three ways we can roll an odd number one, three
and five.
But if we want an odd number and not a one, we can't include this one value in our set one, three
and five.
So the only way to satisfy B and not A or an odd number and not one is with a three or a five.
So there's two possibilities.
So we put two over here and then in the center this is A and B both occurring.
So how many ways are there to get a one and an odd number?
Well, there's only one way because there's only one way to get a one, but one is itself an odd number.
So there's exactly one and only one way that we can roll a one and an odd number, and that is to roll
a one itself.
So we fill in our Venn diagram and then if we want to use this Venn diagram to determine the probability
that A or B occurs, going back to our addition rule formula, we want the probability of A or B, the
probability of A or B, all we have to do is follow the formula and look at the probability of a well,
the probability of a If we look at the entire circle of a here, there's one possibility that a occurs.
In other words, we add up all of the values that are inside the circle A so zero plus one or one.
There's one possibility that A occurs.
We know there's six total possibilities when we roll a six sided die one time.
So the probability of a occurring is one over six, the probability of event B occurring.
We add up all the values inside the circle.
B that's one plus two or three.
So we say plus three over six, but then we subtract this intersectional probability.
That means subtracting the intersection of the Venn diagram.
And inside the intersection here we have this value of one.
So we subtract one over six and we get one over six plus three over six minus one over six or one plus
three minus one all over six is equal to.
Three over six or one half.
So the probability of A or B occurring in this scenario is one half.
The probability of rolling a one or an odd number is one half.
And that makes sense because when we look back at our chart here, the probability that we get a one
or an odd number just intuitively is going to be satisfied when we roll a one or a three or a five,
which is exactly half of the possibilities when we roll the six sided die one time.
So specifically plugging into this addition rule formula here, the probability of getting a one or.
An odd number is one over six plus.
Three over six minus.
And then instead of our intersectional probability being zero like it always is with mutually exclusive
events, when we have overlapping events, we subtract the intersection, which in our case is one over
six.
So we get minus one over six is equal to one plus three minus one over six or three over six.
Three over six is equal to one half.
And so the probability of getting a one or an odd number is one half the other way.
We can calculate this probability when we have overlapping events.
It's just to add up all the numbers in our Venn diagram so we can just add zero plus one plus two to
get three.
Three is our number of matching events.
We divide that by six, the number of total events, and so we get three over six or one half the same
value we got here.
In other words, two ways to calculate the probability here.
If we've built out the Venn diagram, we can just add all the numbers in the Venn diagram.
Zero plus one plus two is three three divided by six.
We get our probability of one half or we can add up all the values in Circle A to get the probability
of a add up all the values in circle B to get the probability of B, and then subtract out the intersectional
probability to eliminate that overlap.
And we'll get to the same answer three over six or one half, 50% chance of rolling a one or an odd
number.
So again, the whole idea here with this addition rule and this union versus intersection concept is
just that we have to pay special attention to or be careful with this intersectional probability right
here, because when we're calculating the probability of one event or another occurring, we don't want
to double count events that occur in both sets over here with this overlapping probability, one occurred
in both set A and set B, And so if we don't subtract out this intersectional probability, if we don't
subtract out this value, we will double count it as part of event A and part of event B and will overstate
our probability.
Going back to our chart here, if we just say that the probability of event A is one over six.
So this event right here is one over six and the probability of event B is three over 6123 matching
outcomes divided by six possible outcomes.
So plus three over six.
And we just add those two probabilities together we get four over six or two thirds or a 67% chance
that we get one or an odd number.
But of course we know already that that's not true.
The probability of getting one or an odd number is one half.
It's 50%, not two thirds or 67%.
This calculation where we just add the two individual probabilities together, double counts this value
of one in both probabilities.
And so we overstate the probability.
We have to account for the fact that one occurs in both sets.
And we need to make sure that we don't double count it and instead that we only count it one time eliminating
the overlap here or taking out that intersectional probability where we subtract out the probability
of A and B occurring, we subtract out the probability of A and B occurring.
We eliminate that double counting and we end up with an actually accurate probability for this scenario
of overlapping events.
So that's how we use the addition rule to calculate the probability that event A or event B occurs by
using this concept of union and intersection and translating that into this visual idea of the Venn
diagram where we can clearly show mutually exclusive events with non overlapping circles or overlapping
events with these overlapping circles in our Venn diagram.