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Independent vs. Dependent Events

Independent vs. Dependent Events

Author: Jonathan Osters

Identify if an event is independent or dependent and determine its corresponding probability.

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Source: Die, public domain Dice, public domain Playing Cards; Public Domain: Quarters, Marbles, and Deck of Cards created by Author

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In this tutorial, you're going to learn about the difference between events that we can consider independent versus dependent events, events that we can't consider independent of each other. So, once we define one, we'll be able to figure out what the other one means. Let's take a look.

Let's start off on an example. Suppose that you did an experiment that consisted of two parts. You selected a card from among the 52 cards in a deck, and you rolled a die. So, you don't know what card you're going to pick, and you don't know what number you're going to get on a die. So, we're not sure here, and we're not sure here.

But what if I gave you an additional piece of information? What if I told you for sure-- for sure-- that the card that you were going to pick was the four of clubs? Would that provide any additional information about what the die was going to be when you rolled it? It wouldn't, and so we can call these events independent.

Independent events means that knowing the outcome of the first event doesn't affect the probabilities for the second event. So, rolling two dice would be considered independent events. Knowing what happened with the first die doesn't affect what's going to occur on the second die. Or flipping two coins-- knowing that this first coin came up heads doesn't affect the probability that the second one will come up heads or tails. You could also talk about selecting two marbles from a jar, or drawing two cards from a deck.

Now, these have asterisks by them because there's a caveat here. There's a warning that I need to give you. They're only going to be considered independent if you actually put the first thing back before you select the second thing.

You have to put the first marble back and mix up the marbles in the jar before you select the second one, or you have to put this first card back and shuffle the deck before you select the second one. The first marble or card has to be replaced. If they're replaced, you can consider the two draws from the jar or two draws from the deck to be independent. The probability of red won't change.

Let's give an example here. Consider this selection here. The probability that you get a spade on the first draw is one-fourth. One out of every four cards are spades. So, suppose that a spade is drawn the first time and not replaced. So, suppose I picked the 10 of spades here, and it's gone. If the two events of drawing were independent, then the probability of selecting a spade wouldn't have changed for the second draw.

But here, there are only 51 cards remaining because I removed the 10 of spades, 12 of which are spades. 12 spades out of 51 cards is not one out of four. It's not the same as it was on the first draw. So knowing that we got a spade on the first draw affects the probability of a spade on the second draw. These two events-- these two draws from the deck-- are not independent. They're dependent events.

Let's look at that same scenario, except have the card replaced. The first draw has a probability of one-fourth of being a spade. So, suppose that you pull the 10 of spades again, but then you put it back. What's the probability of a spade on the second draw? Well, you know what? It's the same 52 cards and the same 13 spades, so it's still one-fourth.

The probability of a spade on the second draw isn't affected by knowing that you got a spade on the first draw. We could have known that we got a diamond on the first draw, or a face card on the first draw, and it wouldn't affect the probability of drawing a spade.

So, let's consider another example. Suppose you draw one card from the deck. What's the probability that it's a six? Well, there are four 6's out 52 cards, so one-thirteenth. But suppose I peek at the card and told you it was a club. What's the probability now of it being a six, if I told you that it was a club?

Well, here are the clubs, and these are the sixes among the clubs. And so, there's one 6 remaining out of the 13 clubs. One-thirteenth is still the probability. It didn't change. And so, knowing that the suit was a club doesn't change the probability of a six. That means that the events-- selecting a club, and selecting a six-- are considered independent. So, six and club are independent when they're done on the same draw.

Events that aren't independent are called dependent events. With dependent events, knowing what happened on the first event affects the probability for the second event. So, suppose that you roll a die and cover it so you don't know what's rolled. What's the probability that you rolled an odd number? Well, that's not too hard. Three of the faces are odd, and so, three out of six is one-half.

However, suppose that I peeked at it and told you it's a high number, and we know that one, two, and three are considered low, and four, five, and six are considered high. What's the probability now of it being odd? Well, we know it's not low, so now we're limited to the high numbers. The probability of an odd now is one-third. Knowing that it was high changed the probability of it being an odd number.

So, when you add this new layer of knowledge, the probability gets affected. That means that the events in question-- high number and odd number-- are dependent on a die.

And so, to recap, independent events are events where the knowledge of what happened on the first trial doesn't change the probability that the second event will occur. A lot of the times, independent events occur when sampling with replacement, whereas they create dependent events if you sample without replacement.

And so, we talked about independent events, dependent events, and then sampling both with and without replacement. Sampling with replacement often creates independent events. Sampling without replacement often creates dependent events.

Good luck, and we'll see you next time.

Terms to Know
Dependent Events

Two events where knowing whether the first event occurred affects the probability of the second event occurring.

Independent Events

Two events where knowing whether the first event occurred does not affect the probability of the second event occurring.

Sampling with replacement

A sampling method where each selected item is replaced into the sampling frame before the next trial. Using this method, an item can be selected more than once.

Sampling without replacement

A sampling method where each selected item is not replaced into the sampling frame before the next trial. Using this method, an item can only be selected once, and the probabilities of particular events may change as subsequent trials are performed.