by Jason Amores Sumpter

In this video, we're going to introduce the rule of multiplication, which is also sometimes called the and roll. Now the rule of multiplication, as its name implies, is going to involve multiplication. And the rule of multiplication is also sometimes called the product role or the end of roll. Now we'll explain why it's called the and roll here very shortly later in this video. But the reason it's called the product role is because the product is the answer to a multiplication problem. And so, by saying product roll, it's really just, uh, implying that multiplication is involved. And so the rule of multiplication, the product rule and the and rule are all referring to the same exact thing. And they basically say that the probability for multiple independent events, or the probability for greater than or equal to two independent events to occur together is calculated by multiplying the chances that each event occurs alone. And so, for example, the probability that two coins, coin number one and coin number two both land on tails. We need to take the probability of one coin landing on tails alone and multiply it by the probability of another coin landing on tails alone. And when we do that, we get one half times one half, which is equal to 1/4. And so again, the reason that this rule of multiplication is called the and rule of times is because the word and is usually used to refer to two events occurring together. So, for instance, coin one and coin too, both landing on tails. And so if we take a look at our image down below, over here on the left hand side, we're showing you, uh, these probability of flipping two coins at once. And so if we flip the first coin, we know that there's a 50% chance of landing on heads and a 50% chance of landing on tails or one half probability of landing on tails. But again, we're focusing here for the sake of this example, the probability that two coins both land on tails so we only have it labeled for the tails. We know that there's a 50% chance that coin number one will land on tails when we flip it. Then when we flip coin number two, it's a completely independent event. So the result of coin one is not going to impact the result of 10.2, So there's still a 50% chance that coin to will land on heads. I'm sorry on tails or on heads on DSO. What we need to do is if we want the probability that both coins will land on tails together. This is where we need to implement the rule of multiplication. And so we take the probability of one coin landing on tails alone, which again the first coined the probability that it lands on tails alone is one half. And then we take the probability that the second coin lands on tails alone, which is still one half. And if we want the probability that these two coins will both land on tails together, then we multiply these two probabilities one half times one half, which gives us 1/4. And so there's a 1/4 probability that both coins will land on tails. And that's exactly what we see here in this square that there is a 1/ probability that both coins will land on tails. Now over here on the right hand side, what we're showing you is another application of the multiplication rule by looking at this particular example, which says if hetero zegas parental pea plants have to offspring, what is the probability they will both be green? And so what we need to realize is that one fertilization event is independent from another fertilization event, and Soto have to offspring. We are really talking about two fertilization events that are independent of one another. And so, of course, when we take a look at this planet square, we're looking at a parent number one, which is hetero zegas parent number two, which is also hetero zegas. We fill in the pundit square and this is the probability that we see. And the probability that one offspring will be green, we know is one out of four total possibilities. 1/4 eso. We can even fill in 1/4 probability that it will be green, that one offspring will be green. But this is looking at the probability that two offspring will be green, not the probability that one offspring will be green. What's the probability that one off the first offspring will be green, and the second offspring will also be green, also known as the and roll here and so to do that, what we need to do or consider is the probability that the first offspring is green. So probability of offspring number one being green is 1/4. And because each fertilization event is independent, the probability that offspring number two is green is also 1/4. And so if we want the probability that both offspring are going to be green, then we need toe implement the multiplication rules. So we take the probability off one independent event and multiply it by the probability of another independent event. So 1/4 times 1/4 is actually one 16. And so the probability that both offspring are going to be green is 1/16 on. That is the answer to this example. Problem right here. And so this year concludes our introduction to the rule of multiplication, the product rule and the and rule, and we'll be able to get some practice applying these concepts as we move forward in our course. And then we'll talk about the rule of addition. So I'll see you all in our next video

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