Mathematics is full of counterintuitive facts and problems that push your mind to new areas and challenge your cognitive versatility.
The next nine facts are just a few of those puzzles; problems that have a habit of sparking arguments and debates among even some of the smartest students.
While they may seem controversial, they're all rock solid fact.
Several of them are paradoxes and idiosyncrasies of probability. Others play your instincts against your logic. If you're looking for a mathematical way to impress your friends and beguile your enemies, here's a good place to start.
1. The most popular gamble strategy is a losing one.
It is known as The Gambler's Ruin.
The gambler raises his bet when he wins.
Since the problem is called Gambler's Ruin it should be obvious - he's screwed.
This is an actual betting strategy.
2. The Gin and Tonic Problem
- Say you've got a glass of gin and a glass of tonic.
- You take a 40ml shot of the tonic and pour it into the gin.
- You take a 40ml shot of tonic/gin mixture and pour it into the tonic.
Is there now more gin in the tonic or tonic in the gin?
Here's what the problem looks like:
The solution:
There is exactly the same amount of tonic in the gin cup as there is gin in the tonic cup.
- You added 40ml of the tonic to the gin cup.
- Then you took 40ml of that mixture and poured it into the tonic cup.
- That second shot contained X milliliters of tonic and 40-X milliliters of gin. So there are 40-X milliliters of gin in the tonic cup.
- Since you poured 40ml of tonic into the gin cup, but took back X milliliters, there are 40-X milliliters of tonic left in the gin cup.
3. Streaks of heads or tails during coin flips are actually really common.
Some coin flip probability:
- Everybody knows when you flip a fair coin, there is a 50% chance it comes up heads and 50% chance it comes up tails.
- However, that's just in the abstract. If you flip a 1000 coins, it's highly unlikely that you get exactly 500 heads and exactly 500 tails.
- Here's the question: if I flip a coin 1000 times, how many streaks of 5 heads in a row should I expect? How many streaks of 10 heads?
Notice the pattern?
Using that, we can figure out how many streaks, say 7 in a row we can expect over 1000 flips.
4. Should you buy Flood or Volcano insurance?
You move to an area that has floods with a 100-year recurrence interval and volcanic eruptions with a 50-year recurrence interval.
This means that each year there's a 1 in 100 chance of a flood event and 1 in 50 chance of volcanic event.
How long will you have to live here before you are 50% sure you'll see one of these events?
Here's how to approach it:
- The best way to solve this problem is by looking at the annual probability that neither event happens.
- Since bot a volcano and a flood would wreak havoc on your property, either event happening is bad.
- Once we figure out the probability of not getting hit by a natural disaster in a given year, we can figure out how long it's going to take before you're pretty sure it's going to happen.
First, figure out the probability of neither event happening in a year:
Here's how we figure out our annual odds:
We can extrapolate for each new year and each annual risk. Notice how the probability of a disaster occurring is steadily rising:
The break point - the point where the probability you've been hit with either disaster exceeds the probability you haven't - is at year 23:
How can you use this idea?
- Consider the idea of purchasing the flood insurance.
- You can calculate your expected payout and compared the expected accumulated cost to figure out if purchasing the insurance is worth it - or if you should just save the money on your own.
- This is basic example of what actuaries and insurance companies do to calculate your rates.
5. The problem with false-positive disease screening tests.
There's a disease screening test that you plan to take to see if you're infected.
- Let's say there's a disease that impacts 1% of people in a population.
- The test always finds the disease but has a 5% false positive rate.
- You get tested and it's positive for the disease.
- How severely should you freak out?
You probably shouldn't freak out.
Here's why. Let's look at 10,000 people who take the test.
So what does this mean?
- The test said that a total 595 people of the 10,000 had the disease.
- Of those 595 people, only 100 (17%) people were true positives.
- A whopping 495 (83%) of them were false positives.
- You should probably get a second opinion.
6. If you want to guess someone's job based on one attribute, you're probably wrong.
I want you to guess Mark's career.
The only thing I tell you about Mark is that he follows politics very, very closely.
Knowing that, is it more likely that Mark is a political scientist or a bus driver?
Since he's into politics, you think it's probably more likely he's a political scientist. Right?
For starters, the Bureau of Labor Statistics says that there are 652,590 bus drivers in the US.
The BLS estimates there are 5,750 political scientists.
So for every political scientist there are 114 bus drivers.
So the question really becomes: Do you think that more than 1 in every 114 bus drivers follows politics very, very closely?
If the answer is YES, then you would say that it is more likely Mark is a bus driver than a political scientist.
If you'd say NO, you would say that it's more likely he is a political scientist.
Let's look at the math showing why.
We wan't to figure out the probabilities Mark is in either job given that he is 'political'.
So in this, we're estimating whether Mark is either a political scientist or a bus driver. We ignore any other possible career.
Let's get our initial probabilities given that we know he's either a political scientist or a bus driver based on the BLS occupation numbers.
There are 114 bus drivers for every political scientist. If we didn't know Mark was political, this would be our estimate:
PROBABILITY MARK IS A BUS DRIVER:
PROBABILITY MARK IS A POLITICAL SCIENTIST:
Now we wan't to estimate the likelihood that someone is political given they are in either career.
So, the final answers:
The probability Mark is a Political Scientist, given that I tell you he's interested in politics is:
The probability Mark is a Bus Driver, given that I tell you he's interested in politics is:
So what does this mean?
The probability Mark is a Bus Driver, given that I tell you he's political is 1,140/(1,140+99)=92%
which means the probability he's a Political Scientist is 8%.
7. Prosecutors are using misleading stats to send people to jail.
As a member of a jury, you have to understand the use of statistics in the court.
- You're on the jury for assault trial for an incident that took place at the Yankee Stadium.
- The key witness said he saw the assailant wearing a Boston Red Sox jersey. That's the key evidence linking the accused to the crime.
- Yankee Stadium management told the police that according to their security footage, 9% fans at the game wore Boston jerseys and 91% wore Yankee jerseys.
- In the test replicating the conditions of the incident, police say the witness was able to distinguish a Boston jersey with a Yankee jersey with 80% accuracy.
- Do you convict?
- First off, it seems like witness is pretty confident the guy was wearing Boston jersey.
- The witness also performed well on the exam. An 80 is a good score.
- There weren't even that many Boston fans at the game, so it's even more likely it's him.
- Boston fans are notoriously uncouth.
All of that (except the last one) is the wrong to think about it. In fact, there's a good chance he's not guilty. Here's why...
We're going to figure out the probability that the shirt witness said he saw, was actually a Boston shirt using the table below:
First we plug the possibilities into this probability table:
Then we solve for the joint probabilities, the ones in the middle. These are 'AND' statements:
The blue box is the probability the witness "saw" a Red Sox jersey:
The red box is the probability the assailant wore a Red Sox jersey and the witness saw a Red Sox jersey:
This is actually a huge issue, too.
- Prosecutors have been accused of overstepping their bounds and misleading jurors by misusing statistics.
- The math and statistical community has alleged the criminal justice system abuses the fact that most Americans don't have a strong intuitive sense of statistical reasoning.
- While this was a flippant example, in real courts potentially innocent people are sent to prison because the jury didn't understand confidence intervals or margin of error.
8. Not all Monopoly spaces are equally likely.
- Monopoly is a game that combines both chance and skill.
- Statisticians have figured out the chance part.
- After years of work, they have successfully sucked the last bit of fun out of America's most dangerous game. (It is a known FACT that Monopoly brings out the worst in every person!)
One way to figure out the odds on landing on a space is through 'Markov Chains':
It's all about the probability you move to another state with Markov Chains:
With Monopoly, every roll of the dice is one such switch.
- We can make the same sort of chart - and find the same sort of probabilities - for the Monopoly Board.
For example, someone who rolls the dice on the 'Go' has these probabilities for what space they go to next:
There's a reason not all the spaces are equal.
- Each space on the board has probabilities like this for where the next roll can take you.
- This is complicated by the Chance card and the Go To Jail space, but those can be factored in.
- In fact, they're the main reason all spaces in Monopoly are not equal!
- By building a large Markov Chain model, statisticians were able to find the probabilities of landing on each space of the board.
The 3 most popular - and dangerous - spaces on the Monopoly board?
9. The rational numbers are infinite, but countable.
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
Obviously, there are an infinite number of rational numbers between, say, 0 and 1. However, are they countably infinite? That is, can they be ordered in such a way that you can systematically count them up?
George Cantor discovered a way to order the numbers so they can be counted.
First, you go along the diagonals:
As you go along, cross out repeated fractions that can be simplified:
Now you have an ordering that can be tied to the counting numbers. So it is a countably infinite set:
Congratulations! You didn't quit!
If you missed the previous article, here it is: 12 Most Controversial Math Facts That People Refuse To Believe Are True.
This article was adapted from Walter Hickey / BI.