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Independence is one of the most difficult mathematical concepts for a high school student to grasp. This is largely because it is a concept as opposed to a simple formula. The Quadratic Formula looks scary, for example, but ultimately it's just a formula. Independence is not just a formula (though one certainly exists), but a concept, and one that can be somewhat counter-intuitive at that. However, it's nothing of which one should be too frightened, and is actually simpler than it seems.

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Two events are independent if each does not influence the probability of the other happening. Remember that this is the mathematical definition of the word "independent", not the general definition. Two mathematically independent events may have absolutely nothing to do with each other, or they may have plenty in common. The only thing that matters is that they do not have any direct influence over each other. Let us look at two examples - one where independence is inherently obvious, one where it isn't - to see exactly how the idea works, and to serve as a gateway into the difference between independent and dependent events.

Let's say you have a shirt drawer with 15 shirts in it. 3 of those shirts are blue. Two obviously independent events in this case are "the odds that today is Saturday" and "the odds that you select a blue shirt at random from your shirt drawer". On the surface, the color of your shirt and the day of the week are clearly unrelated. Time doesn't care if your shirt is blue. The odds of today being Saturday are 1/7 regardless of the color of your shirt, and the odds of your shirt being blue are 1/5 (simplified from 3/15) regardless of what day it is. Easy enough, right? Let's look at something a bit harder.

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Take a standard deck of 52 playing cards. Let's imagine we're drawing a card at random. (Go ahead and actually do so, if you like!) The probability that we draw an ace is 4/52, or 1/13, because there are 4 aces in a standard deck. The probability of drawing a club is 13/52, or 1/4, because there are 13 clubs in a standard deck. What are the odds of drawing a club given you have drawn an ace? Well, there are 4 aces in a deck, so the probability is going to be out of 4. You have an ace of clubs, an ace of diamonds, an ace of spades and an ace of hearts. One of the four aces is a club, so the odds of drawing a club are 1/4 - exactly what they were before we placed the ace restriction! Think this is a coincidence? Try it in the other direction. What are the odds of drawing an ace given you have drawn a club? There are 13 clubs in a deck, so the probability is going to be out of 13. Within the club suit, you have an ace, a 2, a 3, a 4, a 5, a 6, a 7, an 8, a 9, a 10, a jack, a queen and a king. One of those cards is an ace, so the odds of drawing that ace are 1/13 - again, exactly what they were before we placed the club restriction. We can thus say that "the odds of drawing an ace from a standard deck of playing cards" and "the odds of drawing a club from a standard deck of playing cards" are independent. They did not influence each other at all, despite the fact that they both case from the same deck of playing cards, and that there's even one card that would satisfy both probabilities!

So, if two events can be independent despite being quite related in concept, what's the difference between independent and dependent events? As independent events deal with events that do not alter each other's odds, dependent events deal with events that do. A quick and easy example can be found in rolling a fair die. Take your two events to be "rolling a three" and "rolling an odd number", which are obviously related. The odds of rolling a three are 1/6, and the odds of rolling an odd number are 3/6, or 1/2. What are the odds of rolling a three given we have rolled an odd number? Well, there are three odd numbers on a die: 1, 3 and 5. Of those three outcomes, one is our desired outcome, the 3, so the odds are 1/3. This is different from the original odds of 1/6. Rolling an odd number changed our odds of rolling a 3. Now look at it in the other direction. What are the odds of rolling an odd number given we rolled a 3? Well, 3 is odd, right? So the odds are 1/1, or 100%, which has changed from the odds of 1/2 we had earlier when we weren't given that we had rolled a 3. Each of these two events influenced the odds of the other, and so they are dependent.

It is also possible to have two seemingly unrelated events can be dependent as well. Without delving into more numbers, think of two events like "watching a football game today" and "going out partying with friends tonight". Choosing to watch a football game could make you more likely to go out drinking with friends because it will give you something to talk about, or it may make you less likely because you eat a lot while you watch the game, or because the game you want to watch is at the time your friends want to go out. Similarly, choosing to go out with friends could make you less likely to watch a football game because the only game you want to see is on when you'll be out, or it could make you more likely because your friends are going to go somewhere with a TV that is showing a game you want to watch but couldn't at home. The two events, despite being seemingly unrelated on the surface, are very much dependent events.

Ultimately, the key to handling independent and dependent events is to think of them only in terms of their mathematical definitions. Math does not care if two events seem closely related (drawing an ace, drawing a club) or rather unrelated (watching a football game, going out with friends), it only cares whether or not they influence the other's probability. Thinking of independence in this manner is the major mental hurdle to learning how independent and dependent events work, but it's not too hard to do with some effort, and once you've done it successfully, independence suddenly isn't so scary anymore.

Now go forth and think about whether or not "reading this essay to the end" and "memorizing the Quadratic Formula" are independent.

Two events are independent if each does not influence the probability of the other happening. Remember that this is the mathematical definition of the word "independent", not the general definition. Two mathematically independent events may have absolutely nothing to do with each other, or they may have plenty in common. The only thing that matters is that they do not have any direct influence over each other. Let us look at two examples - one where independence is inherently obvious, one where it isn't - to see exactly how the idea works, and to serve as a gateway into the difference between independent and dependent events.

Let's say you have a shirt drawer with 15 shirts in it. 3 of those shirts are blue. Two obviously independent events in this case are "the odds that today is Saturday" and "the odds that you select a blue shirt at random from your shirt drawer". On the surface, the color of your shirt and the day of the week are clearly unrelated. Time doesn't care if your shirt is blue. The odds of today being Saturday are 1/7 regardless of the color of your shirt, and the odds of your shirt being blue are 1/5 (simplified from 3/15) regardless of what day it is. Easy enough, right? Let's look at something a bit harder.

Take a standard deck of 52 playing cards. Let's imagine we're drawing a card at random. (Go ahead and actually do so, if you like!) The probability that we draw an ace is 4/52, or 1/13, because there are 4 aces in a standard deck. The probability of drawing a club is 13/52, or 1/4, because there are 13 clubs in a standard deck. What are the odds of drawing a club given you have drawn an ace? Well, there are 4 aces in a deck, so the probability is going to be out of 4. You have an ace of clubs, an ace of diamonds, an ace of spades and an ace of hearts. One of the four aces is a club, so the odds of drawing a club are 1/4 - exactly what they were before we placed the ace restriction! Think this is a coincidence? Try it in the other direction. What are the odds of drawing an ace given you have drawn a club? There are 13 clubs in a deck, so the probability is going to be out of 13. Within the club suit, you have an ace, a 2, a 3, a 4, a 5, a 6, a 7, an 8, a 9, a 10, a jack, a queen and a king. One of those cards is an ace, so the odds of drawing that ace are 1/13 - again, exactly what they were before we placed the club restriction. We can thus say that "the odds of drawing an ace from a standard deck of playing cards" and "the odds of drawing a club from a standard deck of playing cards" are independent. They did not influence each other at all, despite the fact that they both case from the same deck of playing cards, and that there's even one card that would satisfy both probabilities!

So, if two events can be independent despite being quite related in concept, what's the difference between independent and dependent events? As independent events deal with events that do not alter each other's odds, dependent events deal with events that do. A quick and easy example can be found in rolling a fair die. Take your two events to be "rolling a three" and "rolling an odd number", which are obviously related. The odds of rolling a three are 1/6, and the odds of rolling an odd number are 3/6, or 1/2. What are the odds of rolling a three given we have rolled an odd number? Well, there are three odd numbers on a die: 1, 3 and 5. Of those three outcomes, one is our desired outcome, the 3, so the odds are 1/3. This is different from the original odds of 1/6. Rolling an odd number changed our odds of rolling a 3. Now look at it in the other direction. What are the odds of rolling an odd number given we rolled a 3? Well, 3 is odd, right? So the odds are 1/1, or 100%, which has changed from the odds of 1/2 we had earlier when we weren't given that we had rolled a 3. Each of these two events influenced the odds of the other, and so they are dependent.

It is also possible to have two seemingly unrelated events can be dependent as well. Without delving into more numbers, think of two events like "watching a football game today" and "going out partying with friends tonight". Choosing to watch a football game could make you more likely to go out drinking with friends because it will give you something to talk about, or it may make you less likely because you eat a lot while you watch the game, or because the game you want to watch is at the time your friends want to go out. Similarly, choosing to go out with friends could make you less likely to watch a football game because the only game you want to see is on when you'll be out, or it could make you more likely because your friends are going to go somewhere with a TV that is showing a game you want to watch but couldn't at home. The two events, despite being seemingly unrelated on the surface, are very much dependent events.

Ultimately, the key to handling independent and dependent events is to think of them only in terms of their mathematical definitions. Math does not care if two events seem closely related (drawing an ace, drawing a club) or rather unrelated (watching a football game, going out with friends), it only cares whether or not they influence the other's probability. Thinking of independence in this manner is the major mental hurdle to learning how independent and dependent events work, but it's not too hard to do with some effort, and once you've done it successfully, independence suddenly isn't so scary anymore.

Now go forth and think about whether or not "reading this essay to the end" and "memorizing the Quadratic Formula" are independent.