When discussing the two addition rules for probability we first need to start with
defining mutually exclusive events. And mutually exclusive events, sometimes called disjoint events,
are events that cannot occur at the same time. Let's look at this from a visual perspective, here
is a Venn diagram and the red circle represents the probability of event A, and the green circle
represents the probability of event B, and the rest of the area, the blue color in the square,
represents any other probabilities in the sample space. We know the total for probability is equal
to 1, so the total area in our diagram is equal to 1. And in our diagram A and B have
no overlapping areas, meaning they cannot occur at the same time so they are mutually exclusive.
For instance probability of A could represent rolling a 3 on a die, probability of B could
represent rolling 6 on a die, and the rest of the blue area would represent rolling a
one, two, four, or five. Here is another diagram, but probability of event A and probability of event B
have an overlapping area, so these are not mutually exclusive, so they can't occur at the same time. For
instance let's say that you're picking a movie to watch from your personal collection, probability of
A could represent a movie that won best actress award, probability of B could represent a movie
that won best actor award. The overlapping area probability A and B would represent a movie that
won both Best Actress and Best Actor Awards, and the rest of the area would then be a movie that
did not win either of these awards. Now to addition rules. The first addition rule of probability is
that when 2 events, such as A and B are mutually exclusive, the probability that event A or B will
occur, is probability of A or B equals probability of A plus probability of B. For instance going back
to our die roll example, where event A is rolling a 3, and event B is rolling a 6, so again, they are
mutually exclusive. In this case we would use this rule, so probability of A or B equals, 1 over 6, the
probability of A, plus 1 over 6, the probability of B. And this ends up being 1 over 3, or 0.333
or 33.3%. This rule can extend beyond 2 events. Let's say you're in a raffle with a total of 92
participants who have one raffle ticket each. And there will be eight $100 gift cards awarded, 15 free oil
changes awarded, and 18 t-shirts awarded, with no repeat winners. So, what is the probability that
you win something? I will also fill in this diagram as we go through the math we can assign the $100
gift cards as event A, the oil changes as event B, and the t-shirt as event C, and since there
are no repeat winners these events are mutually exclusive. Our equation is probability of A or B
or C, equals probability of A, plus probability of B, plus probability of C. With 92 participants, the
probability of A equals 8 over 92, probability of B equals 15 over 92, and the probability of
C equals 18 over 92. Add these up and we get 41 over 92, which equals 0.445,
or a 44.5% probability of winning something. Now to the second addition
rule, which applies when 2 events, such as A and B, are not mutually exclusive, so they can
occur at the same time. And it is probability of A or B equals the probability of A plus the
probability of B, minus the probability of A and B. Going back to our actress and actor award diagram,
if we assign movies that won best actress as event A, and it has a probability of 0.18, and we assign
movies that won best actor award is event B, and it has a probability of 0.11, and the overlapping
area of A and B has a probability of 0.03. Using our equation, we have the probability of A or B,
equals 0.18 plus 0.11, minus 0.03, which is 0.26 or a
26% probability of a movie that won Best Actress or Best Actor. Alright my friends, hopefully
this video helped you out, I do have more videos right there for you, till next time, I am outta here.