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In the previous video, we've looked at mutually exclusive probability and non mutually exclusive probability,

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this video, we're going to look at conditional probability.

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Now, the events that occur can be considered independent if the likelihood of one event occurring does

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not change the likelihood of another event occurring.

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Perfect example of this is a coin toss or a roll of the dice.

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Getting a heads or rolling a set number will not affect the probability of getting the same event again

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or a different event.

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Every time you flip a coin or roll the dice, every outcome is equally as likely, no matter what the

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outcome was from the previous event.

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Now events can be considered dependent if they are the opposite.

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When one event occurs, it changes the probability of other events occurring.

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A deck of cards is a perfect example of this, if we pull out any card at random, we have a 50/50 chance

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of it being a black card or a red card.

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And we've previously calculated this before.

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So the probability of pulling out a red card is 26 over 52.

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So 26 red cards in this deck, 52 cards in total.

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We work it out is point five.

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And we do the same thing for black is point five as well.

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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.

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So if you pull out any card from a full deck, let's say we pull out a three of hearts.

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It was a 50/50 chance is a red card or a black card.

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And if we put it back and pull out a different card and this time we pull out a black seven of clubs,

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we still have a 50/50 chance of being black or red.

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Now, these are considered independent events.

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So what happens if we continue this experiment without replacing the last card we've just pulled out?

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So in the last example, we just had put out the seven of clubs.

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So what happens if now we don't replace this card, so we leave it pulled out?

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We're going to have to change the whole probability of getting a red or a black card.

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And we can calculate this again using basic probability.

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So the probability of pulling out a red card now has changed.

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It is two.

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It is now 26 over 50 to minus one.

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So we pulled out one card.

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So there is no longer 52 cards in the deck.

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So we subtract one from it.

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So we get a slightly higher probability than 50 percent of pulling out a red card.

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And if we look at the black now, we have twenty six minus one because we pulled out one black card

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already and we have one last card in the deck.

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So we have 50 to minus one.

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So this is equal to this point for nine.

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So a slightly lower chance of than 50 50 or pulling out a black card.

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So this is considered dependent events.

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Since we've already had one event occurred, the the probability of another event occurring has changed.

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So if we have a closer look at dependent events, the probability of event A and then event B occurring,

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if they are dependent, can be described using this equation here.

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So the probability of A and then event B occurring is equal to the probability of A and then the probability

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of event B happening, given that A has occurred.

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So this bar here represents the word given the probability of drawing out a red card followed by a black

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card without replacing the cards after drawn can be expressed using this equation up here.

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And it can be written as this equation.

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The probability of pulling out a red card and then a black card is equal to the initial probability

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of pulling a red card and then the probability of pulling on a black card.

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Given that a red card has been drawn and we can fill in the numbers from the previous slide and come

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up with this probability here.

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So the probability of pulling out a red card initially is just the twenty six fifty two.

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So that's just point five.

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And then the probability of pulling out a black card, given that a red card has been previously drawn,

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is this twenty six over 51.

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So we multiply them together, we get a probability of this point two five five so we can mathematically

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express conditional probability using this term here.

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So again, this is the probability of an event, a occurring given the event B has occurred, is equal

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to the probability of A and B occurring, divided by the probability of B occurring.

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So these terms and equations here are commonly referred to as conditional probability.

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This term up here is called the joint probability because it's a joint probability of A and B occurring.

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And this is usually referred to as the marginal probability.

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Now to work out whether an event is dependent or independent, it follows a few simple rules.

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So independent events follow this equation here.

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The probability of A and B occurring is just equal to the probability of eight times the probability

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of B, there's no dependence on either of the other ones.

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So if we look at the conditional probability, the conditional probability of A occurring given that

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B has occurred, is just the probability of a..

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So there's no dependence on B and likewise B right around the other way.

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The probability of Event B occurring, given that A has occurred, is still just equal to the probability

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of B, again, there's no dependence on.

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And you can easily say this relationship up here from this total relationship up here.

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So if the probability of A and B occurring is just a multiple of the probabilities, the probability

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of A given that B occurred is just going to be the probability of eight times the probability, B,

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divided by the probability of B, leaving the probability of A..

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So that's just what this equation here is expressing.

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Now, let's look at another example to help explain the concept here we have a number of shapes displayed

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on screen and they're different colors.

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So if we pull out a shape at random, we can work out a few different probabilities.

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So the probability that we pull out a circle is three of right.

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So there's eight shapes and three of them are circles.

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So it's pretty simple to calculate that and we can do the same thing for the square.

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So to pull out a square where there's five different squares and there's eight shapes in total.

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So basic probability holds, we can also look at joint probability.

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So the probability point at a circle and its screen is going to be equal to one over eight.

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So there's only one green circle in the set here.

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Now we can use conditional probability to calculate a few other properties.

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So what is the probability that if we pull out a circle that it is green?

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Well, we can use conditional probability and we can fill in this equation here and it can be calculated

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by the probability of pulling out of green and circle divided, the probability of pulling out a circle.

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So you fill in these values up here into the equation.

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We end up here.

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We end up with one of our eight divided by three or eight, which is just equal to one third.

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So if we pull out a circle, there's a one in three chance that it'll be green.

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And we can easily verify this from looking at this set up and we can calculate it directly.

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We can say there's three circles and only one of them is green.

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So one, seven, three.

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So this is a fairly trivial example, but we can always come back to a basic probability and calculate

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it directly from a set by inspection.