This tutorial covers independent and dependent events. First, we'll tell you what the definitions are. Then we'll look at sampling and how independent and dependent events come out in sampling. And finally, we'll run through a set of examples.
To start off, independent events are two or more events where knowing that one event has occurred does not change the probability of the other occurring. The key here is this does not. And because the not is the important part, I'm going to leave that circled.
Now for example, if we looking at the population of the US and selecting a person from that, would the events race and gender be independent? Now let's think about it. Does knowing someone is male change the probability of them being a particular race? The answer is no because the genders are pretty evenly distributed throughout the races. So those two events, gender and race, are independent.
On the other hand, we have dependent events. And dependent events are two or more events where knowing that one has occurred does change the probability of the other. So that's that does. And so the changing of the probability is the key part there.
So if instead we were looking at, let's say, gender and height, are those two events independent or are they dependent? Well, knowing that someone is male does change the probability of their height. It changes whether or not they're going to be tall or short. While there are some short males, males are taller than females. So if you know someone is a male, then they are more likely to be taller. So that does affect the probability of being taller. So gender and height are then dependent events.
Here, when looking at sampling, the ways we sample can affect whether or not the events are dependent or independent. So for this first part here, we're sampling with replacement. So that means that we're taking items from the population. We're taking elements. They could be people, or they could be things.
Once we've taken an element out, if we put it back into the population before drawing the next one again, then we are sampling with replacement. Each time, we're starting from that same total population and pulling our sample out from that same total because we've always put back in what we took the time before. So in this case, what I take the first time and then I put it back in and what I take the second time, those don't change anything. The probability stays the same. I have the same probability of drawing something on the first pick as I do the second as I do the 100th. Because every time, we're putting everything back into the population. So this, sampling with replacement, the events are going to be independent because the probability isn't going to change from one pick to the next.
On the other hand, sampling without replacement, that's when we're taking out an element. And then once we've taken it out, it stays out. Then we take out another element. And then once we take it out, it stays out. We are not replacing it. It's without replacement. So as I take things out, my population that I have to choose from is getting smaller and smaller and smaller.
So here, because we're not returning it, the probability changes with every draw. Let's say I had 10 items, and 3 of them are red. And the first time I pull something out, now I only have nine things. So my probability of getting a red now depends on what I had pulled out.
And then when I pull something out again, the probability of what I pick on that third draw depends on what I pulled out the first two times. So because it depends on it, because the probabilities change as we go through, that's a dependent event. So here, our selections are dependent. Whereas here, our selections are independent.
Now let's look at a couple of examples and decide whether the events are independent or dependent. So for starters, drawing an ace and drawing another ace. Well, that depends on whether or not we're putting the cards back in before we draw the second one or not.
So here, with an independent event, if the two cards are returned, so you pull out a card, you return it, and then you pull out another card. That would be independent. Here, if you pull out the card and it stays out, then you pull out another card, then that's going to be dependent.
So this one could go either way. Oops. I wrote on it instead. So I'm going to leave that in the middle. And it would be independent if we were replacing. And it would be dependent if we were not replacing.
Now our next example, rolling a 2 and then rolling 3 using a die. Here, when I roll a die the first time and I roll the die the second time, those don't change anything. Knowing that I rolled a 2 doesn't affect what can come next. So those are independent events.
Here, if I say that I know a roll is odd, and I know a roll was a 3. Are those independent or dependent? Well, once I know a roll is odd, then it can only be a 1, a 2, or a 5. So the probability of being a 3 then is 1/3. If I don't know anything about what happened, the probability of being a 3 is 1 out of 6. So there, the probability changes because I knew it was odd. So that's going to be dependent.
And our last example is rolling a 5 and drawing a 5. So if I rolled a 5 on a die, and then what's my chance of pulling a 5 out? Knowing I rolled a 5 doesn't change anything about my card draw. So that is an independent event.
Independent and dependent events are something that's very important for later tutorials about probability. So make sure that you understand the key differences between the two. This has been your tutorial on independent and dependent events.