In the previous video, we've looked at mutually exclusive probability and non mutually exclusive probability,
this video, we're going to look at conditional probability.
Now, the events that occur can be considered independent if the likelihood of one event occurring does
not change the likelihood of another event occurring.
Perfect example of this is a coin toss or a roll of the dice.
Getting a heads or rolling a set number will not affect the probability of getting the same event again
or a different event.
Every time you flip a coin or roll the dice, every outcome is equally as likely, no matter what the
outcome was from the previous event.
Now events can be considered dependent if they are the opposite.
When one event occurs, it changes the probability of other events occurring.
A deck of cards is a perfect example of this, if we pull out any card at random, we have a 50/50 chance
of it being a black card or a red card.
And we've previously calculated this before.
So the probability of pulling out a red card is 26 over 52.
So 26 red cards in this deck, 52 cards in total.
We work it out is point five.
And we do the same thing for black is point five as well.
So if we pull out any card at random, we have a 50/50 chance of it being a black card or a red card.
So if you pull out any card from a full deck, let's say we pull out a three of hearts.
It was a 50/50 chance is a red card or a black card.
And if we put it back and pull out a different card and this time we pull out a black seven of clubs,
we still have a 50/50 chance of being black or red.
Now, these are considered independent events.
So what happens if we continue this experiment without replacing the last card we've just pulled out?
So in the last example, we just had put out the seven of clubs.
So what happens if now we don't replace this card, so we leave it pulled out?
We're going to have to change the whole probability of getting a red or a black card.
And we can calculate this again using basic probability.
So the probability of pulling out a red card now has changed.
It is two.
It is now 26 over 50 to minus one.
So we pulled out one card.
So there is no longer 52 cards in the deck.
So we subtract one from it.
So we get a slightly higher probability than 50 percent of pulling out a red card.
And if we look at the black now, we have twenty six minus one because we pulled out one black card
already and we have one last card in the deck.
So we have 50 to minus one.
So this is equal to this point for nine.
So a slightly lower chance of than 50 50 or pulling out a black card.
So this is considered dependent events.
Since we've already had one event occurred, the the probability of another event occurring has changed.
So if we have a closer look at dependent events, the probability of event A and then event B occurring,
if they are dependent, can be described using this equation here.
So the probability of A and then event B occurring is equal to the probability of A and then the probability
of event B happening, given that A has occurred.
So this bar here represents the word given the probability of drawing out a red card followed by a black
card without replacing the cards after drawn can be expressed using this equation up here.
And it can be written as this equation.
The probability of pulling out a red card and then a black card is equal to the initial probability
of pulling a red card and then the probability of pulling on a black card.
Given that a red card has been drawn and we can fill in the numbers from the previous slide and come
up with this probability here.
So the probability of pulling out a red card initially is just the twenty six fifty two.
So that's just point five.
And then the probability of pulling out a black card, given that a red card has been previously drawn,
is this twenty six over 51.
So we multiply them together, we get a probability of this point two five five so we can mathematically
express conditional probability using this term here.
So again, this is the probability of an event, a occurring given the event B has occurred, is equal
to the probability of A and B occurring, divided by the probability of B occurring.
So these terms and equations here are commonly referred to as conditional probability.
This term up here is called the joint probability because it's a joint probability of A and B occurring.
And this is usually referred to as the marginal probability.
Now to work out whether an event is dependent or independent, it follows a few simple rules.
So independent events follow this equation here.
The probability of A and B occurring is just equal to the probability of eight times the probability
of B, there's no dependence on either of the other ones.
So if we look at the conditional probability, the conditional probability of A occurring given that
B has occurred, is just the probability of a..
So there's no dependence on B and likewise B right around the other way.
The probability of Event B occurring, given that A has occurred, is still just equal to the probability
of B, again, there's no dependence on.
And you can easily say this relationship up here from this total relationship up here.
So if the probability of A and B occurring is just a multiple of the probabilities, the probability
of A given that B occurred is just going to be the probability of eight times the probability, B,
divided by the probability of B, leaving the probability of A..
So that's just what this equation here is expressing.
Now, let's look at another example to help explain the concept here we have a number of shapes displayed
on screen and they're different colors.
So if we pull out a shape at random, we can work out a few different probabilities.
So the probability that we pull out a circle is three of right.
So there's eight shapes and three of them are circles.
So it's pretty simple to calculate that and we can do the same thing for the square.
So to pull out a square where there's five different squares and there's eight shapes in total.
So basic probability holds, we can also look at joint probability.
So the probability point at a circle and its screen is going to be equal to one over eight.
So there's only one green circle in the set here.
Now we can use conditional probability to calculate a few other properties.
So what is the probability that if we pull out a circle that it is green?
Well, we can use conditional probability and we can fill in this equation here and it can be calculated
by the probability of pulling out of green and circle divided, the probability of pulling out a circle.
So you fill in these values up here into the equation.
We end up here.
We end up with one of our eight divided by three or eight, which is just equal to one third.
So if we pull out a circle, there's a one in three chance that it'll be green.
And we can easily verify this from looking at this set up and we can calculate it directly.
We can say there's three circles and only one of them is green.
So one, seven, three.
So this is a fairly trivial example, but we can always come back to a basic probability and calculate
it directly from a set by inspection.