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So previously we looked at the addition rule for probability, and now we want to look at the multiplication

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rule for probability.

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When we talk about the multiplication rule, we have to consider both independent and dependent events.

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We're going to use the multiplication rule in two different ways, depending on whether or not our events

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are independent or dependent.

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But first, let's just start with the multiplication rule itself.

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The multiplication rule tells us that the probability of events A and B both occurring is equivalent

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to the product of the probability that event occurs and the probability that event B occurs.

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And we've just written this multiplication rule in two different ways.

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The first way using this intersection notation and the second way using the word.

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And remember that this intersection probability symbol represents this idea of and probability.

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And so to write this as the probability of a intersection, B or the probability of A and B is really

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just expressing the same idea, just written in two different ways.

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Let's start looking at this multiplication rule when we use it to apply just to independent events.

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So independent events are events that don't affect each other.

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Think about two separate coin flips.

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The result of the first coin flip doesn't affect the result of the second coin flip or rolling a die,

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picking it up and rolling it again.

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Those two DI roles are independent of each other.

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What I get on the first roll doesn't affect what I get on the second roll.

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So if we have independent events and we want to find the probability that they both occur, we call

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that a joint occurrence and we multiply the individual probabilities of the two events.

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In other words, let's just take the classic example where I'm flipping a coin two times and I want

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to know the probability that both coin flips come up as tails.

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That means that event A is getting tails on the first flip and event B is getting tails on the second

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flip.

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So I could write that probability as the probability of tails and the intersection of that with tails.

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I could also express this as the probability of tails.

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And tales, or sometimes we'll see this written as the probability of just tails tails.

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But we're trying to indicate the same joint occurrence in each of these three representations.

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And no matter which representation we use, our multiplication rule formula tells us that the probability

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that we get tails on the first coin flip is, of course, just one half.

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Because if I flip a coin, assuming it's a fair coin, there's a 50% chance I'll get heads and a 50%

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chance I'll get tails.

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So I have a one in two chance of flipping tails.

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So the probability of event A is one half.

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And then I need to multiply that by the probability of event B, Well, event B is getting tails on

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the second coin flip.

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And that second coin flip, really, if I pick up the coin again after flipping it the first time,

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I'm now holding that coin in my hand.

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And it doesn't matter whether I flipped heads or tails previously, I now have a coin held in my hand.

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I'm about to flip it and it has no knowledge of how it previously landed.

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I have exactly a one and two chance of getting heads and a one and two chance of getting tails on that

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second flip.

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So the probability of getting tails on the second flip is also one half.

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So the probability of event A is one half.

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The probability of event B is one half.

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I multiply these together and I get one half times one half is one fourth.

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So the probability of flipping tails two times in a row is one fourth.

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We could also think about this coin flip scenario by imagining all of the possibilities when we flip

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a coin two times.

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So if we flip a coin two times, there are four equally likely outcomes.

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I could get heads and then heads.

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I could get heads and then tails.

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I could get tails and then heads or I could get tails.

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And then tails.

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These are the four possible scenarios, the four equally likely outcomes of flipping a coin two times

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in a row.

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And so because I'm looking for the specific scenario of tails and then tails, if I take this back to

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my addition rule formula, I can say that I have one matching outcome with the scenario I'm looking

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for tails, tails.

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So there's one matching outcome and there are four total outcomes one, two, three, four total outcomes.

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So the probability that I get tails, tails when I flip a coin four times is one in four or one fourth

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or a 25% chance.

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And of course this rule doesn't just work with coin flips.

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Let's take an example where maybe we'll say that the probability that a factory produces a defective

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part is 2%, and we want to know the probability of picking three defective parts in a row if we're

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just picking parts randomly off of the assembly line, Well, we could write that here as the probability

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of a defective part and then another defective part and then another defective part.

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In other words, we could almost think about this as events A, B, and C where event A, event B and

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event C are all picking a defective part.

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Well, we said that the probability of the factory producing a defective part was 2%, which means that

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the probability of event a being choosing a defective part is going to be 2%.

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Or we could write that as two over 100.

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Then the probability of choosing another defective part is again 2%.

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So multiply by 2% or two over 100 and the probability of choosing a third defective part is also 2%

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or two over 100, because at least as far as we can tell, these events are independent.

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I'm choosing completely randomly a part off of the assembly line.

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This factory produces defective parts at a 2% rate.

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So the chance that any one part that I choose is defective is 2%.

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Of course, we have to be careful here because we might think that these events are independent, but

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let's say we pick three parts in a row and maybe the group of employees who's working in that particular

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hour when we choose parts, actually produces defective parts at a higher rate than an employee group

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who works a different shift in the same factory.

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And we have to take that into account or there's maybe multiple assembly lines in the same factory and

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one line produces more defective parts than the other.

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So we have to be really careful about the independence of these events.

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But assuming we know these events are independent, this is how we would actually do the math.

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And when we multiply here, we get two times.

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Two times two is eight and then 100.

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Cubed is 1 million.

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If we simplify eight over 1 million, we get one over 125,000, which means there's a one in 125,000

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chance that randomly choosing three parts off of the assembly line out of the factory results in three

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defective parts.

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So this is how we calculate probability for a joint occurrence when we have independent events and we

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want to know the probability that those independent events all occur.

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But when we have dependent events, we use this set of formulas.

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So this set of formulas for dependent events, this is the same multiplication rule formula, except

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that you can see that we've changed one detail, which is that we have changed this probability of B

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piece of the multiplication formula into this expression here, The probability of B given that event

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A has already occurred.

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That's what this expression means here.

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When we separate B and A with this line, we're saying the probability that event B happens given that

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A has already occurred.

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So we're looking for the probability that this happens, given that this has already occurred, regardless

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of what A and B actually are, the probability that the first thing happens given that the second thing

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has already occurred.

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The classic example here is pulling cards out of a standard 52 card deck.

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For instance, let's say we want to know the probability of pulling a king and then another king.

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So a king and then another king from a deck of 52 cards.

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And remember, in a deck of 52 cards, there are four suits, and each suit contains 13 cards from two

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all the way up through ten.

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And then Jack, Queen, King, ace.

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So 13 cards in each suit for suits, 52 cards in total.

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And we want to know the probability that we pull one king and then another KING Well, we could do this

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as an independent events problem, assuming that after we pull the first king, we put it back into

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the deck before pulling another card.

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If we pull the first king and we leave it outside the deck and then pull another card, then we're talking

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about dependent events, because the probability of pulling a king on that second pull is dependent

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on what we polled in the first poll.

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Because if you imagine if we start out with 52 cards and four of those 52 cards are kings, if we pull

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a card from this 52 card deck and we don't replace it, then when we go to pull a second card, we're

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only going to have 51 cards to pull from because one of the cards is missing.

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We already pulled it in our first poll.

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There's only 51 cards left in the deck from which to poll.

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And so right there automatically, we know that we're looking at dependent events.

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If we don't replace that first card and we can think about that idea of dependent events, because this

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second event depends on this first event, it is changed by it is affected by whatever we do with this

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first event.

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Whereas if I flip a coin two times, that second flip is not affected in any way by the first flip.

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But if I'm pulling cards from a deck and I pull the card and don't replace it, then that second poll

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is going to be affected by the fact that I pulled a card already.

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So let's assume that's what we've done here.

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We're looking for the probability of getting a king on our first poll and a king on our second poll.

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And we are not replacing the first card after we pull it from the deck.

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Well, in that case, the probability of event A, the probability that we get a king on our first poll

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is going to be four in 52 because there are four kings in the deck, 52 cards in the deck in total.

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So there are four matching outcomes and 52 total outcomes.

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So probability of event A is.

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For over 52.

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Now, the probability of event B, given that A has already happened in our example, means the probability

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that we pull a king, given that we already pulled a king and didn't put it back in the deck.

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Well, if we already pulled a king, our deck is down one card.

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There's only 51 cards left in total.

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And if we already pull the king, then there are only three kings left in that deck.

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Which means the probability that we pull a king in that second poll is three out of 51 three matching

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outcomes out of 51 total outcomes.

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So we can multiply here by three over 51 three over 51 is the probability of event B given that A has

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already happened or the probability we pull a king, given that we've already pulled a king and we didn't

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put it back.

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So we can then simplify the math here, four over 52 is the same as one over 13 and three over 51 is

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the same as one over 17.

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So one over 13 times one over 17 is one over 221.

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There's a one in 221 chance that we pull a king and then another king, assuming we didn't replace the

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first king after we pulled it.

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So this is our classic dependent events example.

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What this shows us then is that we can use these two formulas to prove whether or not events are independent

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or dependent, because these two formulas here are identical.

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We talked about this earlier.

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These two formulas are identical, except that we're saying here P of B, the probability of B is getting

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replaced with the probability of B given A Other than these two things here, the formulas are identical,

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which means that if we can show that the probability of B is equal to the probability of B, given that

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A has already happened.

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In other words, if we can show that these two values are equivalent, then we know that the events

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have to be independent, because what this is saying is that the probability of Event B happening,

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given that A has already happened, is the same as the probability that B happens.

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In other words, event A happening had no effect on the probability of event B happening.

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If event A already happened, event B is totally unaffected.

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And if it's totally unaffected, that means that B must be independent from A because A has no effect

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on B.

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So if we can show that this relationship holds, then the events have to be independent.

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Think back to our coin flip example from earlier.

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If we say that event A is flipping heads on one flip and that event B is flipping heads.

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On a second flip, we know that the probability of event B flipping heads on this coin flip is one half,

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so the probability of event B is one half.

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And this probability here, the probability of event B given event A we can think about as the idea

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that we flipped the coin once and we got heads on that first flip.

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Now we're looking at event B, we've picked up the coin again and we're looking at the probability of

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flipping heads given that we flipped heads on the previous coin flip.

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Well, this second coin flip is totally unaffected by whether we got heads or tails on the last flip.

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So the probability of B given a meaning, the probability of flipping heads, given that we flipped

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heads last time, is still just one half.

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So we can say the probability of B given A is one half.

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And so then because these two values are equivalent, the resulting logical conclusion here is that

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A and B.

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Must be independent.

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Of one another.

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They must be independent events because we showed that this equation was true here, that these two

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probabilities were equal and therefore that the probability of B was totally unaffected by event A already

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happening.

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And therefore we know that events A and B must be independent.

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So this is another check we can use for confirming whether two different events are independent or dependent

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events.

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But this is how we use the multiplication rule to calculate the probability that two events both occur,

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the probability of event A and event B both occurring if the events are independent.

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We use this formula here.

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We just multiply the probabilities of the individual events.

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If the events are dependent, then we're calculating conditional probability because one event is conditional

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on the other, one event is dependent on the other.

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And so we replace this probability of event B expression with the probability of event B Given that

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event A has already happened in order to account for the dependence of one event on another, the probability

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of one event is conditional on the other event.

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And so we modify our formula in this way to calculate the probability that multiple events are occurring.