The coin and dice examples in the previous video are all examples of mutually exclusive events.
This just means that only a single event can occur at the same time.
So if we have a look at the single coin toss event example, we have a set of events of a heads or tails
being shown on the up face so we can even get a heads or tails is impossible for us to flip a coin and
get a heads and tails at the same time.
So the event, heads and tails are mutually exclusive.
If we also have a look at the single roll example, we can have a set of the possible events being the
one, two, three, four, five or six dots showing up or down here, but is impossible to roll the
dice with multiple faces shown on the up face.
At the same time, it is impossible to roll a three and a four at the same time.
Only a single one of each of these events occur at a single time.
So this can be expressed mathematically, so given a set of possible events, A and B. So the set of
events and B basic probability says the probability of getting an A or A B or so any of the complete
set is just the sum of the probabilities and it has to equal one.
We have to get one event out of the total set.
Now, events A and B occurring simultaneously is impossible, so the probability of getting A and B
equals zero, then events A and B are mutually exclusive is impossible to get an event A and B occurring
at the same time.
But events do not have to be mutually exclusive.
It is possible to get different events occurring at the same time.
So here we have a static pack of playing cards.
So we have numbers from Ace all the way up to King, and we have four different suits.
Now, it is possible to pull out a hot card from this example, from this deck.
It is also possible to pull out a king from this deck.
And it's also possible to pull out a king of hearts, so the probability of pulling out a king and a
heart does not equal zero.
So this is a non mutually exclusive event.
In the event of pulling out a heart and pulling out a king is not mutually exclusive.
So let's have a look in a bit more mathematical detail, so, for example, if we wanted to work out
the probability of pulling out a hot card, we can see that there are 13 hot cards in the stick and
the deck is made up of 52 cards.
Therefore, the probability of pulling out a hot card is just 13, over 52.
And we can do the same thing for the king cards.
So we can easily see there are four cards and there's 52 cards in the deck.
So the probability of pulling out a king is for over 52.
So now say we want to work out the probability of pulling out a card or a king card.
Now, when, Wolf, we're looking at the mutually exclusive events, we could just add the two probabilities
together so we could do the probability of a heart or the probability of the king and the probabilities
together.
So we could be doing this for non mutually exclusive events.
This is not a valid and we can easily see from this sort of Venn diagram up here, we can see that this
set of events and this set of events intersect each other.
So they have a common event that can occur at the same time.
So if we add this probability up and this probability up together, we can see that we've included this
probability of the king of hearts happening live, included it twice.
So this is not going to give us the correct probability.
So the correct way to calculate the probability of pulling out a heart or a king is going to be the
sum of the probabilities of the heart and king minus the common intersection, which is going to be
the probability of pulling out of king and heart at the same time so we can put numbers under this.
So we have probability of pulling out of King is the four and a 50 to the probability of pulling out
a heart is 13 of 52 and the probability of pulling out of king and a heart at the same time or simply
say that's only a single event.
So that's just one over 52.
So if we evaluate this equation, we get 16, 52.
So that's a probability of pulling out a heart or a king.
And we can quickly verify this because we can see there's 13 cards here and then we have 14, 15, 16.
So there's 16 out of 52 ways of pulling out a king or a heart card.
So to summarize what we've talked about in this video, mutually exclusive events means that the probability
of Event A and Event B occurring at the same time is zero.
So if the probability is zero, then the events are mutually exclusive.
If the probability is not zero, then the events are not mutually exclusive.
It is possible for Event A and Event B to occur at the same time.
Now, the probability of non mutually exclusive events can be written in this equation down here, so
the probability of A or the probability a B is just the sum of the probability of the individual event,
minus the probability of A and B occurring.
So we could quickly see that if the events are mutually exclusive, then this is just going to be this
part just going to be equal to zero.
And we're going to be left with our original equation as a probability of A and A probability, B or
B, but if they're not mutually exclusive, then we have to take that into account by subtracting a
single common the by subtracting the common overlap between the events.