Conditional probability is a concept that jumps into the game when talking about experiments about which we know certain facts.

"Probability of finding a treasure in a cave that's somewhere in a mountain, assuming we are 100% sure we will find or already found that cave".

When we know certain facts, the experiment changes. We don't have the previous experiment anymore.

"Probability of finding the treasure".

And that's it.

That makes me think: "Okay, what is this conditional probability all about then? What's the purpose of defining such a concept?".

It seems useless.

It just sets a mathematical relationship between two different experiments.

Every mathematical relationship is a bridge that allows some playing. But still, I have two questions:

  • Why is this relationship useful, if it is useful at all?
  • Is there a way to visualize the relationship graphically?
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    $\begingroup$ I think a more interesting question is the opposite: why is the concept of non-conditional probability useful? If you notice, in real life, practically all probabilities are conditioned by many other random variables. Just a few and rare cases are those where you have completely independent random variables. $\endgroup$ – Masacroso Jul 21 '16 at 13:29

I almost laughed when you said it seemed useless. As Masacroso says, it's the opposite, since almost everything in the real world has probability that's highly dependent on our knowledge of relevant facts. Let's consider some examples to understand this.

  • What's the probability that there's a rainstorm in my location in one hour? Well, I could take into account no information whatsoever and simply, say, integrate the total area on Earth that's seen rain with respect to a time variable that goes back millennia (as some kind of a priori probability that speaks to all times and places on the planet). Or, I could notice that at my location currently the sky is completely clear of clouds and the sun's shining brightly in order to guess there will be no rain in the next hour.
  • Say you want to search for valuable shipwrecks in the ocean. You could assume an a priori probability of it being located anywhere in the ocean with uniform likelihood, or you could use historical facts about its disappearance to narrow the shipwreck down to a nontrivial distribution over say a dozen square kilometers.
  • If you want to count cards in a card game to gain an advantage, the whole point is to keep track of the current state of the deck (what cards it has and doesn't have) in order to have more accurate predictions about what could be in your competitors' hands.

These are all examples of real world conditional probability. Supercomputers predict future weather conditions based on those conditions leading up to the current moment at various locations, marine salvage has been aided with the help of Bayesian search theory, and counting cards uses information to condition probability evaluations.

Nate Silver is a statistician that is famous for correctly predicting 49/50 state outcomes in the 2008 U.S. presidential election, which was bumped up to 50/50 in 2012. His models tracked polling data to make its predictions. Why wouldn't the probability of a candidate winning an election not be conditioned according to current polling trends? An a priori probability would be static and wouldn't evolve with new information, it couldn't track the change in opinion as news events happen (speeches, debates, scandals, ads, attacks, etc).

If anything, it seems in real world situations that it's unconditioned probabilities that are useless. As in the first bullet points, a weather prediction that's blind to your current time, location and weather conditions is useless. Finding a shipwreck by searching the whole ocean when you know perfectly well it resides in a certain dozen square kilometers is useless. Trying to guess election outcomes from a position of complete ignorance about how the campaigns have unfolded and how people's attitudes have evolved is useless.

Imagine a world where criminal cases were decided solely based on the a priori probability of a defendant's guilt, refusing to hear any specific evidence for or against her which would condition the probabilities. That would be far worse than useless, it would be a deranged, dystopian world.

Finally, the way to visualize conditional probabilities is with Venn Diagrams.

$\hskip 2in$ venn diagram

Say circle $A$ encompasses all outcomes of a certain type. If the diagram represents probability by, say, area, you could divide the area of $A$ by the area of the whole diagram to obtain the a priori probability that an outcome is in $A$. But what if you also just so happen to know something about the outcome, namely you know that it's in $B$? Then you would divide the area of $A\cap B$ by the area of $B$. This could be a very different probability.

In fact, given any two probabilities $0<p<1$ and $0\le q\le 1$, show that there are dependent (say discrete) random variables $X$ and $Y$ and values $x$ and $y$ for which $\mathrm{P}(X=x)=p$ but $\mathrm{P}(X=x\mid Y=y)=q$. The fact that the a priori probability doesn't necessarily tell us jack squat compared to the conditional one shows just how useless the unconditioned probability can be.

  • $\begingroup$ Why does the probability (trust ratio?) in conditional probability involve the probability of event B, which we know that already happened, being it the condition? Also I don't see what events-set $A\cap B$ is in your three examples. Take the first: "A: There is a rainstorm in the next hour AND B: the sky is clear?". Back to the case I mentioned: "Find treasure knowing I found the cave" (What is the purpose of involving $P(B)$: "Find the cave" in the calculations if I already found it?). Ah, I am a million times thankful for your answer. I just want to be able to see this. To be sure about it. $\endgroup$ – Álvaro N. Franz Jul 21 '16 at 16:04
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    $\begingroup$ @ÁlvaroN.Franz To see why $\mathrm{P}(B)$ is relevant to calculating $\mathrm{P}(A\mid B)$, look at the venn diagram. Instead of dividing by the area of the total rectangle, since you know we're in $B$, we can divide by the area of $B$. To know that $B$ has happened but to mathematically do nothing what that information is to feign ignorance and pretend you don't know $B$ at all. Consider for instance, what is the probability of rolling a $5$ with a die, if you know you rolled greater than $4$? You wouldn't say $1/6$th. $\endgroup$ – arctic tern Jul 21 '16 at 16:13
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    $\begingroup$ @ÁlvaroN.Franz To make the difference more apparent, let's consider the probability it will be raining one minute from now in the Sahara desert (event $A$) using the fact that it is raining there right now (event $B$). Most of the time if it's raining one minute it will be raining the next minute too, so $P(A\mid B)$ will be close to $1$, and $P(A)$, $P(B)$, $P(A\cap B)$ are approximately equal. But the fact that it's raining there at all is extremely rare, so these latter three probabilities are close to $0$. $\endgroup$ – arctic tern Jul 21 '16 at 16:38
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    $\begingroup$ In terms of venn diagrams, $P(A\cap B)$ is the area of $A\cap B$ relative to the area of the whole rectangle, whereas $P(A\mid B)$ is the area of $A\cap B$ relative to the area of $B$. $\endgroup$ – arctic tern Jul 21 '16 at 16:40
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    $\begingroup$ $A\mid B$ is not an event (at least not in the same sample space), it is notation that means "$A$ given $B$." If you want you can interpret it as the same set of outcomes as $A\cap B$ but with respect to the potentially smaller sample space $B$ I guess. $\endgroup$ – arctic tern Jul 21 '16 at 17:06

In a non-mathematical, "real-world" sense, conditional probability is useful because every realistic application of probability involves some assumption of priors - that is, background conditions which hold in the world in which the probability is calculated. As such, probabilities are not calculated in a vacuum, but instead calculated given some priors. You normally don't care about absolute probabilities in practice, only probabilities given certain conditions.

Mathematically, conditional probabilities are used everywhere in probability theory, often as an integral part of theorems and formulae that have applications to everything from physics to biology to economics.


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