Wednesday, April 12, 2017

J is for Joint Probability

On an episode of House, Dr. House and his colleagues were discussing (as usual) a difficult case, one in which a woman appeared to have multiple rare diseases. As the doctors argued how impossible that was, House realized that it could absolutely happen, because, as he put it in the show:

I love this quote, because it sums up an important probability concept so succinctly.

You see, his colleagues were coming from the assumption that because the patient had already been hit with an extremely rare event, there was no way another extremely rare event could occur. Almost as though the first rare events protects the patient from more. It's the same when we flip heads on a coin. We expect the next one to be tails.

But this implies that the two events are connected. What if they're independent (which is what House meant when he said "no memory")? It's improbable, but absolutely possible, for two rare events to occur together. In fact, you can determine just how likely this outcome would be.

We call this joint probability: when you determine the probability of two random, independent events, which you determine through multiplication. That is, you multiply the probability of the first event by the probability of the second event, and this tells you the likelihood of a joint event.

Here's a concrete example. Let's use a case where we can easily determine the odds of a joint event without using joint probability - playing cards. You can determine the probability of any configuration of playing cards. If I randomly draw a card from the deck, what is the probability that I will draw a red Queen? We know that there are two red Queens in the deck, so probability is 2/52 or 0.0385.

Here's how we can get that same answer with joint probability. We know that half of the cards in the deck (26/52 or 0.50) are red. We know that 4 of the cards in the deck are Queens (4/52 or 0.0769). If we multiply these two probabilities together, we get 0.0385.

Remember Hillary's 6 coin flips? That's another demonstration of joint probability. In fact, you would find the probability of getting heads 6 times in a row as 0.5^6 (or multiplying 0.5 by itself 6 times). Based on that, we would expect 6 heads in a row to occur about 1.6% of the time.

I'll be talking about set theory later this month, which ties into many of these concepts.

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