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Independent vs. Dependent Events

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Hi. This tutorial covers independent versus dependent events. All right. So consider the two following chance experiments. Experiment one-- draw a card, and note the suit of the card. Set the card aside. Draw a second card. Note the suit of the second card.

So if we were to see that in action, I would take my deck of cards. I'd draw a card. And I'd note the suit. So that is a spade. Set the card aside. Draw a second card. Note the suit of the second card. So that's a diamond. So that would be one trial of experiment one.

Experiment two-- draw a card and note the suit of the card. Put the card back in the deck, then reshuffle. Draw a second card. Note the suit of the second card.

So again, we'll demo that. First card is a club. Now, with experiment two, I need to put that card back in the deck. And I need to reshuffle. So I'll do that, shuffle a couple times. We'll cut it once, and redraw. So club again. So that would end up being one trial of experiment two.

So let A equal the event that the first card is a club. And let B be the event that the second card is a club. So let's try to answer this question. If it is known that event A has occurred, what is the probability of B for each experiment?

So let's start with experiment one. Now, experiment one was where I set the card aside. So now if we know event A already occurred-- event A was the event that the first card was a club-- now, if I want to know the probability of B if the A already occurred, what I would need to do is think about how many clubs would be left in the deck with that card set aside and how many total cards would be in the deck.

So if I already drew a club and put that aside, there would only be 12 cards left. And since that card is not in the deck, there would only be 51 cards left. So that probability would end up being 12 out of 51. And if we figure out what that is as a decimal, 12 divided by 51, that ends up being about 0.235. So about a 23.5% chance of getting a club after we've already drawn a club and set it aside.

Now let's do the same thing for experiment two. So we're assuming that we drew a card. That card was a club. Then we put it back in the deck, reshuffled the deck. And we want to know now what's the probability of getting another club.

So I want to know probability of B now for experiment two. Well, I have all 13 clubs still in the deck. And I have all 52 cards still in the deck. So this ends up being 13 out of 52. And if we do that one as a decimal, that's going to be 13 divided by 52. And that ends up being 0.25, one fourth.

So we can see that these probabilities are slightly different. 23 and 1/2% versus 25%. So those two experiments differed in an important way. It was whether or not we replaced the card. So let's take a look at the definitions of what we call independent events and then what we call dependent events.

We'll start with independent events-- two or more events where knowing one event has occurred does not change the probability of the other events. So when we're dealing with independent events, this is going to be similar to experiment number two. So experiment number two, we put the card back in there, reshuffled. And knowing that we got a club did not affect the probability of getting a club on the second draw. So the key thing there is, does not change the probability.

Now, dependent events is when two or more events, two or more events where knowing one event has occurred does change the probability of the other events. So this is what we saw in experiment number one. So because we knew that that first one was a club, that's going to change that probability. So generally the probability of drawing a club is 13 out of 52. But because we knew that the first event already occurred, that we drew a club, the probability went down to 12 out of 51, which was less than 13 out of 52.

So now, the reason that two of those events were independent and two were dependent was because of the sampling procedure. So sampling with replacement is a method of sampling where an item may be sampled more than once. Sampling with replacement generally produces independent events. So independent events generally occur with sampling with replacement. So again, this was experiment number two where we reshuffled.

And sampling without replacement is a method of sampling where an item may not be sampled more than once, may not be sampled. So sampling without replacement generally produces dependent events. So this is going to be experiment number one. So since you're taking the card out and setting it aside, there's going to be no way of you sampling that card again. So that's going to generally produce dependent events where the occurrence of one event is affected by the occurrence of another event.

All right. Well that has been your tutorial on independent and dependent events. Thanks for watching.