So previously we looked at the addition rule for probability, and now we want to look at the multiplication
rule for probability.
When we talk about the multiplication rule, we have to consider both independent and dependent events.
We're going to use the multiplication rule in two different ways, depending on whether or not our events
are independent or dependent.
But first, let's just start with the multiplication rule itself.
The multiplication rule tells us that the probability of events A and B both occurring is equivalent
to the product of the probability that event occurs and the probability that event B occurs.
And we've just written this multiplication rule in two different ways.
The first way using this intersection notation and the second way using the word.
And remember that this intersection probability symbol represents this idea of and probability.
And so to write this as the probability of a intersection, B or the probability of A and B is really
just expressing the same idea, just written in two different ways.
Let's start looking at this multiplication rule when we use it to apply just to independent events.
So independent events are events that don't affect each other.
Think about two separate coin flips.
The result of the first coin flip doesn't affect the result of the second coin flip or rolling a die,
picking it up and rolling it again.
Those two DI roles are independent of each other.
What I get on the first roll doesn't affect what I get on the second roll.
So if we have independent events and we want to find the probability that they both occur, we call
that a joint occurrence and we multiply the individual probabilities of the two events.
In other words, let's just take the classic example where I'm flipping a coin two times and I want
to know the probability that both coin flips come up as tails.
That means that event A is getting tails on the first flip and event B is getting tails on the second
flip.
So I could write that probability as the probability of tails and the intersection of that with tails.
I could also express this as the probability of tails.
And tales, or sometimes we'll see this written as the probability of just tails tails.
But we're trying to indicate the same joint occurrence in each of these three representations.
And no matter which representation we use, our multiplication rule formula tells us that the probability
that we get tails on the first coin flip is, of course, just one half.
Because if I flip a coin, assuming it's a fair coin, there's a 50% chance I'll get heads and a 50%
chance I'll get tails.
So I have a one in two chance of flipping tails.
So the probability of event A is one half.
And then I need to multiply that by the probability of event B, Well, event B is getting tails on
the second coin flip.
And that second coin flip, really, if I pick up the coin again after flipping it the first time,
I'm now holding that coin in my hand.
And it doesn't matter whether I flipped heads or tails previously, I now have a coin held in my hand.
I'm about to flip it and it has no knowledge of how it previously landed.
I have exactly a one and two chance of getting heads and a one and two chance of getting tails on that
second flip.
So the probability of getting tails on the second flip is also one half.
So the probability of event A is one half.
The probability of event B is one half.
I multiply these together and I get one half times one half is one fourth.
So the probability of flipping tails two times in a row is one fourth.
We could also think about this coin flip scenario by imagining all of the possibilities when we flip
a coin two times.
So if we flip a coin two times, there are four equally likely outcomes.
I could get heads and then heads.
I could get heads and then tails.
I could get tails and then heads or I could get tails.
And then tails.
These are the four possible scenarios, the four equally likely outcomes of flipping a coin two times
in a row.
And so because I'm looking for the specific scenario of tails and then tails, if I take this back to
my addition rule formula, I can say that I have one matching outcome with the scenario I'm looking
for tails, tails.
So there's one matching outcome and there are four total outcomes one, two, three, four total outcomes.
So the probability that I get tails, tails when I flip a coin four times is one in four or one fourth
or a 25% chance.
And of course this rule doesn't just work with coin flips.
Let's take an example where maybe we'll say that the probability that a factory produces a defective
part is 2%, and we want to know the probability of picking three defective parts in a row if we're
just picking parts randomly off of the assembly line, Well, we could write that here as the probability
of a defective part and then another defective part and then another defective part.
In other words, we could almost think about this as events A, B, and C where event A, event B and
event C are all picking a defective part.
Well, we said that the probability of the factory producing a defective part was 2%, which means that
the probability of event a being choosing a defective part is going to be 2%.
Or we could write that as two over 100.
Then the probability of choosing another defective part is again 2%.
So multiply by 2% or two over 100 and the probability of choosing a third defective part is also 2%
or two over 100, because at least as far as we can tell, these events are independent.
I'm choosing completely randomly a part off of the assembly line.
This factory produces defective parts at a 2% rate.
So the chance that any one part that I choose is defective is 2%.
Of course, we have to be careful here because we might think that these events are independent, but
let's say we pick three parts in a row and maybe the group of employees who's working in that particular
hour when we choose parts, actually produces defective parts at a higher rate than an employee group
who works a different shift in the same factory.
And we have to take that into account or there's maybe multiple assembly lines in the same factory and
one line produces more defective parts than the other.
So we have to be really careful about the independence of these events.
But assuming we know these events are independent, this is how we would actually do the math.
And when we multiply here, we get two times.
Two times two is eight and then 100.
Cubed is 1 million.
If we simplify eight over 1 million, we get one over 125,000, which means there's a one in 125,000
chance that randomly choosing three parts off of the assembly line out of the factory results in three
defective parts.
So this is how we calculate probability for a joint occurrence when we have independent events and we
want to know the probability that those independent events all occur.
But when we have dependent events, we use this set of formulas.
So this set of formulas for dependent events, this is the same multiplication rule formula, except
that you can see that we've changed one detail, which is that we have changed this probability of B
piece of the multiplication formula into this expression here, The probability of B given that event
A has already occurred.
That's what this expression means here.
When we separate B and A with this line, we're saying the probability that event B happens given that
A has already occurred.
So we're looking for the probability that this happens, given that this has already occurred, regardless
of what A and B actually are, the probability that the first thing happens given that the second thing
has already occurred.
The classic example here is pulling cards out of a standard 52 card deck.
For instance, let's say we want to know the probability of pulling a king and then another king.
So a king and then another king from a deck of 52 cards.
And remember, in a deck of 52 cards, there are four suits, and each suit contains 13 cards from two
all the way up through ten.
And then Jack, Queen, King, ace.
So 13 cards in each suit for suits, 52 cards in total.
And we want to know the probability that we pull one king and then another KING Well, we could do this
as an independent events problem, assuming that after we pull the first king, we put it back into
the deck before pulling another card.
If we pull the first king and we leave it outside the deck and then pull another card, then we're talking
about dependent events, because the probability of pulling a king on that second pull is dependent
on what we polled in the first poll.
Because if you imagine if we start out with 52 cards and four of those 52 cards are kings, if we pull
a card from this 52 card deck and we don't replace it, then when we go to pull a second card, we're
only going to have 51 cards to pull from because one of the cards is missing.
We already pulled it in our first poll.
There's only 51 cards left in the deck from which to poll.
And so right there automatically, we know that we're looking at dependent events.
If we don't replace that first card and we can think about that idea of dependent events, because this
second event depends on this first event, it is changed by it is affected by whatever we do with this
first event.
Whereas if I flip a coin two times, that second flip is not affected in any way by the first flip.
But if I'm pulling cards from a deck and I pull the card and don't replace it, then that second poll
is going to be affected by the fact that I pulled a card already.
So let's assume that's what we've done here.
We're looking for the probability of getting a king on our first poll and a king on our second poll.
And we are not replacing the first card after we pull it from the deck.
Well, in that case, the probability of event A, the probability that we get a king on our first poll
is going to be four in 52 because there are four kings in the deck, 52 cards in the deck in total.
So there are four matching outcomes and 52 total outcomes.
So probability of event A is.
For over 52.
Now, the probability of event B, given that A has already happened in our example, means the probability
that we pull a king, given that we already pulled a king and didn't put it back in the deck.
Well, if we already pulled a king, our deck is down one card.
There's only 51 cards left in total.
And if we already pull the king, then there are only three kings left in that deck.
Which means the probability that we pull a king in that second poll is three out of 51 three matching
outcomes out of 51 total outcomes.
So we can multiply here by three over 51 three over 51 is the probability of event B given that A has
already happened or the probability we pull a king, given that we've already pulled a king and we didn't
put it back.
So we can then simplify the math here, four over 52 is the same as one over 13 and three over 51 is
the same as one over 17.
So one over 13 times one over 17 is one over 221.
There's a one in 221 chance that we pull a king and then another king, assuming we didn't replace the
first king after we pulled it.
So this is our classic dependent events example.
What this shows us then is that we can use these two formulas to prove whether or not events are independent
or dependent, because these two formulas here are identical.
We talked about this earlier.
These two formulas are identical, except that we're saying here P of B, the probability of B is getting
replaced with the probability of B given A Other than these two things here, the formulas are identical,
which means that if we can show that the probability of B is equal to the probability of B, given that
A has already happened.
In other words, if we can show that these two values are equivalent, then we know that the events
have to be independent, because what this is saying is that the probability of Event B happening,
given that A has already happened, is the same as the probability that B happens.
In other words, event A happening had no effect on the probability of event B happening.
If event A already happened, event B is totally unaffected.
And if it's totally unaffected, that means that B must be independent from A because A has no effect
on B.
So if we can show that this relationship holds, then the events have to be independent.
Think back to our coin flip example from earlier.
If we say that event A is flipping heads on one flip and that event B is flipping heads.
On a second flip, we know that the probability of event B flipping heads on this coin flip is one half,
so the probability of event B is one half.
And this probability here, the probability of event B given event A we can think about as the idea
that we flipped the coin once and we got heads on that first flip.
Now we're looking at event B, we've picked up the coin again and we're looking at the probability of
flipping heads given that we flipped heads on the previous coin flip.
Well, this second coin flip is totally unaffected by whether we got heads or tails on the last flip.
So the probability of B given a meaning, the probability of flipping heads, given that we flipped
heads last time, is still just one half.
So we can say the probability of B given A is one half.
And so then because these two values are equivalent, the resulting logical conclusion here is that
A and B.
Must be independent.
Of one another.
They must be independent events because we showed that this equation was true here, that these two
probabilities were equal and therefore that the probability of B was totally unaffected by event A already
happening.
And therefore we know that events A and B must be independent.
So this is another check we can use for confirming whether two different events are independent or dependent
events.
But this is how we use the multiplication rule to calculate the probability that two events both occur,
the probability of event A and event B both occurring if the events are independent.
We use this formula here.
We just multiply the probabilities of the individual events.
If the events are dependent, then we're calculating conditional probability because one event is conditional
on the other, one event is dependent on the other.
And so we replace this probability of event B expression with the probability of event B Given that
event A has already happened in order to account for the dependence of one event on another, the probability
of one event is conditional on the other event.
And so we modify our formula in this way to calculate the probability that multiple events are occurring.