Conditional Probability
Which is more likely to kill you - a shark or a dog?
According to the Washington Post , in 2001-2013, the United States had an average of one death per year from sharks, versus 28 deaths per year from dog bites. In other words, the average American is 28 times more likely to die from a dog than a shark!
However, in spite of this fact, common sense still says that sharks are much more dangerous than dogs. Why is that? Well, it's because people have way more interactions with dogs than they do with sharks! Sharks are more dangerous, not because they result in more deaths overall, but because the shark is much more likely to kill you if you encounter it in the first place.
This is what is called conditional probability. Conditional probability is the probability that one thing happens, based on some other condition being true. In equation form, we'd write that as P(A|B) — "the probability of A, given B."
In this case, the "given condition" is "encountering a shark" - the probability of dying from a shark is fairly low overall, but the probability of dying from a shark if you've had an encounter with a shark is significantly higher.
Let's demonstrate this with some numbers.
Suppose, hypothetically, that the average American will have 2 interactions with dogs every day.
Since there are about 300 million people in the United States, that works out to 600 million dog encounters happening per day. Over the course of a year, that's 365×600,000,000 = 219 billion human-dog interactions in the U.S. alone! If only 28 of those interactions result in death, then that means that there's only a 0.00000001278% chance of dying from any given dog encounter.
Meanwhile, most people don't usually run into sharks on a daily basis like they do with dogs. In the year 2000, the USA's beach attendance was approximately 118 million . However, the typical beach trip doesn't include a run-in with a shark—the number of shark encounters is probably much lower. Since we don't have statistics for that, let's assume you'll interact with a shark on one out of every four beach trips, giving us 29.5 million shark encounters per year. (That's still probably way too high, but let's go with it.)
1 death per year out of 29.5 million would mean that 0.0000033% of shark encounters end up being fatal.
So, how much more dangerous is it to run into a shark than a dog? If a shark encounter has a 0.0000033% fatality rate, and dog encounters have a 0.00000001278% fatality rate, then a shark encounter is 258 times more likely to kill you than a dog encounter.
Here's a couple examples of how conditional probability might look on your homework:
Example 1
Joe draws a card from a standard deck of playing cards. Given that he drew a face card, what is the probability that he drew a queen?
Since the problem says "given that he drew a face card," we know that this is a conditional probability question. We use the formula P(A|B) = P(A ∩ B)/P(B), or just (number of things where A and B are both true) ÷ (number of things where B is true). "B" is the "given" condition, and "A" is the desired outcome. In this case, B is "face card," and A is "queen."
The top of our fraction in the formula is the number of things where A and B are both true — all the outcomes that match both criteria. In this case, that's any card that is both a face card AND a queen. There are four cards (the four queens) that fit both requirements, so "A and B" = 4.
The bottom of the fraction is the number of things in the "given" category, B. In this case, that's face cards - all the jacks, queens, and kings. There are 12 total face cards, so B = 12.
So, plugging it all in: P(A|B) = 4/12, which simplifies to 1/3.
Example 2
There are 15 marbles in a jar: 5 red, 4 blue, 3 green, and 3 silver. Kamiko draws two marbles from the jar without replacement. If the first marble was red, what is the probability that the second marble is silver?
This problem doesn't say "given," but it's still a conditional probability question - you can tell from the word "if." The given condition is "if the first marble was red," and the desired outcome is "the second marble is silver." So once again, we'll be using the formula P(A|B) = (A and B)÷B.
The top of the fraction (A and B) is the number of items where both conditions are true. In this case, that means pulling a silver marble after a red one was drawn. Taking out a red marble doesn't change the fact that there are 3 silver marbles, so "A and B" = 3.
The bottom of the fraction is the number of items for which the given condition is true, meaning one red marble got drawn. Notice that the problem includes the words "without replacement" - in other words, after taking out the red marble on the first draw, Kamiko didn't put it back. This means there are 14 marbles left in the jar. B = 14.
Plugging it into the formula: P(A|B) = 3/14.
Image Sources:Shark from Alex Steyn on UnsplashDog attacking by Nick Bolton on UnsplashDivers taking picture of shark from Francois on UnsplashExcited puppies from Alvan Nee on UnsplashPlaying cards by Дмитрий Фомин (Dmitry Fomin), CC0, via Wikimedia Commons
