Probability (Multiple events: Independent vs. dependent)

Most calculations dealing with probabilities assume that the probabilities are independent. In other words, they assume that the results of the first event do not affect the other events. Flipping two different coins is a great example of two independent events. We might be superstitious and think that if we get heads on the first coin, we're more likely to get tails on the second flip. But, in fact, each coin flip is totally independent. The odds of getting heads on the first flip is 50% and, no matter what you get on the first flip, the odds of getting heads on the second coin is 50%.

Humans like to see patterns, and we tend to think of independent events as dependent. A classic example is when people are planning their families. When a family has three brothers, they might decide to have another child, thinking, "We have three boys, we're sure to get a girl next time."  But, in fact, the gender of one sibling is completely independent of the gender of any of the other siblings. A family has the same exact probability fo getting a boy on the fourth child as they do on the first child. If you wanted to find the odds of getting several siblings of the same gender, you would just multiply those probabilities together.

Example:

What are the odds of a family having four boys (assuming families have a 50% chance of getting a boy)?

The odds of having a boy are 50% or $\dfrac{1}{2}$.

To find the odds of having four boys, you multiply the probability of having each boy.  These probabilities are independent.  That means that the probability of having a boy is the same each time.

$\eqalign{\text{boy} \times \text{ boy } \times \text{ boy } \times \text{ boy} &= \text{probability of having 4 boys}\\\dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2} \times \dfrac{1}{2}&= \dfrac{1}{16}}$

However, not every probability is independent. Some probabilities are impossible to calculate because we don't know exact how one outcome influences another.  Think about test scores.  When you get a good test score, it probably affects your next test scores. A good test score makes you more confident -- and it also means you know more material. A bad test score means you're struggling, and probably hurts your confidence. Either way, your second test score is not completely independent of your second test score.  However, finding out how those test scores are related is more complicated than just adjusting probabilities (and beyond the scope of this lesson).

But, in other situations, we know exactly how one outcome affects another. In that case, before we multiply the probabilities together, we adjust the second (or third or fourth, etc.) probabilities to show how the prior outcomes influenced them.

Example:

Imagine you have a sock drawer.  In the drawer are 6 white socks, 4 black socks, and 2 gray socks.  You don't pair the socks.  So, when you reach into the drawer, in the dark, in the morning, you pull out whatever sock you reach first.  What are your odds of getting two black socks on the first two grabs?

First, find the probability of getting a black sock on the first grab:

$\dfrac{\text{# of things we care about}}{\text{Total number of things}}=\dfrac{\text{Black socks}}{\text{All socks}}=\dfrac{4}{12}$

Next, find the probability of getting a black sock on the second grab.  What's important to remember here is that we've already removed a black sock from the drawer. These probabilities are dependent. The first grab affects the second grab. The number of black socks in the drawer has changed, so has the total number of socks in the drawer:

$\dfrac{\text{# of things we care about}}{\text{Total number of things}}=\dfrac{\text{Black socks}}{\text{All socks}}=\dfrac{3}{11}$

Finally, multiply the two probabilities together:

$\dfrac{4}{12} \times \dfrac {3}{11} = \dfrac {12}{132}=\dfrac{1}{11}$

A note about "replacement": When doing probability problems, and deciding if events are dependent or not, it's important to look out for the word "replacement." There is no replacement in the problem above. After we pull the first sock, we do not put it back into the drawer (thus, our events are dependent and we adjust the second probability). But, sometimes a problem will tell you to pull a sock out of the drawer, replace it, and then pull a second sock out of the drawer. Those two events would be independent (because, when you do your second grab, you'll still have 4 black socks and 12 total socks in the drawer).  When a problem is "with replacement," the events become independent.  If there is "no replacement," the events are dependent.  

As you can see, the process for finding the probability of multiple events is the same for independent and dependent events.  But, a statistician must remember two important concepts:

1. If events are dependent and you don't know how they affect each other, you can't just pretend the events are independent. These kinds of dependent events are not appropriate for simple probability calculations.
2. If events are dependent and you do know how they affect each other, you must adjust the affected probabilities appropriately for your final probabilities to be correct.