Hi everyone. Today we're going to be talking about the multiplication rule. And the multiplication rule, upon some thought, really kind of makes sense. It's not exceedingly difficult.
There might be some applications that you find a little bit challenging, but the rule itself is not difficult. So again, what we're doing is we're actually talking about what's known as a joint probability. So what that means is two or more events happening at the same time, such as ideal a card, what's the probability it's a jack and a spade.
Okay, so a joint probability is referring to an and statement. That means we want to know what the probability of A and B occurring are in the same experiment. I shouldn't use the word time in the same experiment. Okay? So, I could also be talking about what's the probability that the top card is a jack and the bottom card is a jack.
Right? So, time really isn't an issue. It's really in the same experiment.
I shouldn't have said at the same time. A and B. More commonly, we'll see it written with a logical AND sign or an intersection sign if you're into set theory. And we have the probability of A and B.
So again, if this symbol is unfamiliar to you, just read it as the word AND. Okay? A and B. And if you have a joint probability and you want to use a joint probability, You simply use the multiplication rule.
And the multiplication rule basically means we're going to be multiplying probabilities together. Let's say I had an experiment, and my experiment was... And I have the word roll here, but I mean flip.
I'm flipping a coin three times. What's the probability that I flip a coin three times and I get three heads? So, what's the probability I get three heads? Well, if I flip the coin three times, that means on the first flip, it has to be a head.
The probability of me getting a head on the first flip is 50%, one half. So only one half of the time am I going to get that. Now I have to flip it again, and I have to get another head. So that's also going to occur one half of the time.
So only one half of this one half of the time is that going to happen. So it's going to be one half times one half, which is one fourth. So again, you see multiplication makes sense because what we're saying is, of that 50% of the time, I'm only going to get that second head 50% of that 50%, which makes 25%.
So if I then roll it. A third time I have to get a head again, and so one half of that 25% is going to be three heads. So again, the probability of getting three heads is going to be one half times one half times one half equals one eighth.
So if I pick up a coin, the probability of me flipping three heads in a row is one eighth. Again, On a question like this, where we're potentially dealing in fractions, on a test, I would just want you to leave the answer like this, 1 half times 1 half times 1 half. I wouldn't want you to simplify it.
I wouldn't actually even want you to write 1 eighth. Now, rolling three heads means that, again, I actually have three events going on, and we're actually asking what's the probability of event A? What's the probability of event B?
And what's the probability of event C? And in this case, the events might be fairly obvious and you don't have to go through this step. But when questions become more complicated, if you don't know specifically what the events are, if you haven't really clearly thought about that in your head, you're going to run into trouble.
Trust me. Even on something this simple, I want you to, in your mind or on paper, write down what the events are. Event A is getting ahead on the first flip. Event B is getting ahead on the second flip. And event C is getting ahead on the third flip.
Okay? So the probability of each of those is one-half, and it's one-half times one-half times one-half. So now I want to be a little more formal in...
how I'm writing the multiplication rule and I have the probability of A and B equals the probability of A times and what I have here is the probability of B given A not the probability of B because again we have to think about a conditional probability. What's the probability that I Let's say, what's the probability that I pick a person that is over six feet tall and this person's also more than five feet tall? Well, maybe one-tenth of the people are six feet tall and maybe 90% of the people are five feet tall.
But it's not going to be... 1 tenth by 9 tenths, right? Because if I pick a person who is taller than 6 feet, and maybe that occurs 10% of the time, well, what is the probability he's also taller than 5 feet? Well, it's not 90%. Clearly, if he's taller than 6 feet, he is taller than 5 feet as well.
That conditional probability, the probability that he's taller than 5 feet, given that he's 6 feet, would be 100%. So the probability of me picking someone who's taller than 6 feet and taller than 5 feet, same person, would simply be the probability of me picking someone taller than 6 feet. It would be ridiculous to multiply those two unconditional probabilities together.
If that doesn't make sense, don't worry about it a whole lot. Just understand that the multiplication rule is the probability of A times the probability of B given A. Now, order doesn't matter.
Order, like time, doesn't really matter. The probability of A and B is the same as the probability of B and A. So I actually could have written the formula just as well as The probability of A and B also equals the probability of B times the probability of A given B.
I could, wherever I see A, I could change it to B. Wherever there's B, I could change it to A. Right?
Order doesn't matter. Time doesn't matter. Okay? I know it's tempting to think that.
It's not true. Now, I'm going to talk about a shortcut, though. First of all, this rule is always, always, always true. This is the general multiplication rule. It's always true.
If they're independent, though, we can take a shortcut. If A and B are independent, we know by definition that the probability of B given A equals the probability of B. That's what independence is.
A has no effect on B, right? So if they're independent, the probability of B given A equals the probability of B, and we can simply write the probability of A and B equals the probability of A times the probability of B. So if they're independent, We don't have to worry about the conditional probability.
If they're independent, it's not that this is a different rule for independence. It's like a shortcut because this is still true. Even if they're independent, this rule is still true. It's just that we know this equals this.
So if this is easier to calculate than this, well, let's use this. So we know they're the same number. Which is why with flipping a coin we just see the probability of 1 half times 1 half times 1 half.
Because when you flip a coin, the second flip is independent of the first flip. It doesn't matter if you got a head or a tail on the first flip. That isn't having any say about the second flip, right? So even if you flip 10 heads in a row, the probability that you get a head on that 11th flip is 50%.
Probability of a tail, 50%. So, again, general rule and a specific rule. Basically true if the events are independent.
Okay? So you might want to ask yourself, if you ever get a question like this, probability of A and B, you might want to ask, are A and B independent? Okay?
So, now I've designed a couple of questions, and I want you to post your answers to these. And again, I really want you to post them. Before I make a comment on them, so I have four different events down here We're only going to talk about three at any one time, but we have the first is a diamond So I'm dealing out three cards Person one person to person three this is as with always cards without Replacement what's the pro a is?
the probability that the first person is getting a diamond? The first event is the first person gets a diamond. The second event is the second person gets a diamond. The third defined event is the third person gets a diamond.
And the fourth defined one is what's the probability that the third person gets a king? So now I have two different questions. What's the probability of A and B and C? which is basically saying if I deal out three cards, what's the probability that all of them get diamonds?
Okay. The second question is a little different. It's I deal out three cards, what's the probability that the first two cards are diamonds and the third card is a king? Okay, so don't worry about event C, right? We're not talking about event C at all down here.
Event C isn't mentioned. Event D is not mentioned over here. So don't start thinking about something that's inappropriate, right? We're simply talking about what's the probability that the first card is a diamond, the second card is a diamond, and the third card is a king, right?
Whether or not it's a diamond. So, two questions. When you answer the questions, again, leave them in this form.
Don't give me a decimal answer. Don't simplify. Leave them in a form that looks like this.
That's it for the multiplication rule. I beat the sun today. I'll talk to you later. Bye-bye.