Saturday, February 27, 2010 ~ Have your say...

The Birthday Paradox

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The Birthday Paradox is not a "paradox" in the true sense of the word.  It's more of an "astonishing realization" once people are presented with the proof.

It asks:  How many people are needed in a room before there is a 50% chance that two people share a birthday?  The answer, surprisingly, is only 23.

How can this be?* 

Logically, we assume that this means one individual is asking 23 people if they share the same birthday.

This individual has a one in 365 chance that they share a birthday with one of the others, which is a 0.27% chance.  If I ask 23 people, then it's 23*0.27 = 6.21%. The odds are still pretty low.


With the Birthday Paradox, what we're really looking at is each person asking each other person if they share a birthday.  The odds change dramatically when you put 20+ people in a room and they each ask everyone else if they share a birthday.

We know, a priori, that the magic number to have a 50% chance is 23 people in the room.

Let's do some math...

Remember, a probability of "1" means a 100% chance of the event happening. So, 

"the probability of two people sharing the same birthday" = 1 - (the probability that no one shares a birthday)

In a list of 23 people, comparing the birthday of the first person on the list to the others allows 22 chances for a matching birthday, but comparing every person to all of the others requires a little calculation.

How many pairs can 23 people make?  Since any one person out of 23 can make 22 different pairs, multiply 22 by the total number of people (23) and divide by the amount of people it takes to make a pair (two). This equals 253.

23*22/2 = 253

Now, you have to ask the question “how likely is it for a pair of people to not share a birthday in a year?” A person’s birthday is 1/365 of the year. We can assume someone else can be born on any of the 364 days of a 365 day year, and not share the same birthday. 

This can be written as as:
1-(1/365) = 364/365 = 0.9972

For all 253 pairs to be different, we need to multiply those odds (for example, this is like saying:  "The number of possible outcomes if three dice are thrown is 6 x 6 x 6 = 216":

(.9972)^253 = 0.4995

The chance of a birthday match out of 23 people is then:

1 – 49.95% = 50.05%.

*For these calculations, I am assuming:

1. There are 365 days in a year (ie, not a leap year)
2. There are no twins in the group of 23

I highly recommend the tutorial at Better Explained to brush up on the mathematics behind permutations and combinations - I did!

Observe the graph below - note that to have a 90% chance, you need just over 40 people in the room!
It's amazing how easily we can be fooled - something that seems nearly impossible, upon further investigation, is suddenly highly probable.  But how is the Birthday Paradox useful in our Skeptical Toolkit?  It helps us to digest the actual odds of "coincidences".

"Coincidences" will be reviewed in a future post...  meanwhile, check out this video:

Thursday, February 25, 2010 ~ Have your say...

5 Incredible Science Photos

The Biotechnology and Biological Sciences Research Council - the UK's leading funding agency for academic research and training in the non-clinical life sciences -  recently released the winning images for their science photo contest.  Here are the top 5:



Ant Flexes Its Muscles
This photo of an Asian weaver ant as it sticks to a glass ceiling carrying 100 times its own weight, was first overall in the competition.
(Thomas Endlein, University of Cambridge)


Killer Whale

This photo of a killer whale was the runner-up in the "Concepts" category.
(Emma Foster, University of Leeds)


Soil Bugs

This photo of mites, springtails and insects from a soil sample won in the "Agriculture, Food, Diet and Health" category.
(Felicity Crotty, North Wyke Research and University of Plymouth)

Crow Uses A Bar

A New Caledonian crow uses a stick to pull food from a small hole. This photo won in the "Concepts" category.
(Jolyon Troscianko, University of Birmingham)

Stuck Salmon
This photo won in the "People" category, where a fisherman catches wild salmon in Puget Sound, Washington state.
(Emma Foster, University of Leeds)

Sunday, February 7, 2010 ~ Have your say...

Understanding Scale: The Universe, Atoms, and Homeopathy

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Numbers, especially the very large and very small, can be deceiving.  It's almost impossible for our brains to compute the immense scale in the fields of astronomy or microbiology.  Nevertheless, understanding and being able to visualize the incredibly large scale of our universe is valuable.

Let's take a look at time and distance, and translate smaller units of time into understandibly larger units:

1 second equals 1 second
1 million seconds equals 12 days
1 billion seconds equals 30 years
1 trillion seconds equals 30,000 years!

1 millimeter equals 1 mm
1 million mm equals 1 km
1 billion mm equals 1000 km
1 trillion mm equals 1,000,000 km! (this is like going around the world 25 times)

Here's what $10k, $1 million, and $1 billion looks like:


And here's $1 Trillion:


Understanding these large scales can help us visualize numbers that we might read or hear about.

For example, when you hear that something is "99.999% reliable", it means there is an error rate of 10 out of a million. Using our references from above, that's like being offline for only 10 seconds out of 12 days.  Or, looking at it from the perspective of "distance", you can have a tolerance of 10mm (about the width of your pinky finger) for every kilometer.

The phrase “One part per million” is often used by chemists to measure concentrations of substances. One ppm is like having a presence of 1 second in 12 days. And a part per trillion?Only 1 second every 30,000 years! 

To help humanity visualize the large scale of our universe, the American Museum of Natural History has produced a movie that begins with a view of the Earth's Himalayan Mountains and then zooms out:  showing the orbits of Earth's satellites, the Sun, the Solar System, the extent of humanity's first radio signals, the Milky Way Galaxy, galaxies nearby, distant galaxies, and quasars. Every object in the video has been rendered to scale using the best scientific research available in 2009. The film has similarities to the famous Powers of Ten video, which you can also see below: 


And here is the famous Powers of Ten video:

 
So how does estimating scale and magnitude apply to skepticism and critical thought? 

The pseudoscience of Homeopathy is a great example. 

The "law of infinitesimals" in homeopathy states that dilution increases the curative power of homeopathy medications. This means that a part-per-million solution of a substance is more medicinally powerful than a part-per-thousand solution, which has in turn more curative power than a part-per-hundred solution. In contrast, many of our modern drugs are ineffective in small quantities and the efficacy increases with dosage.

Let us put modern medicine aside and consider the dosages involved in homeopathy.

Homeopathic medicines come in 12x, 24x, 28x dilutions ("28x" means the solution has been diluted 28 times) . If a substance were to be diluted 30 times, this means that there would be one part medicine to one trillion quadrillion parts water (or other inert ingredient).  That's a 1 with 27 zeros, or 1,000,000,000,000,000,000,000,000,000.  Going back to our "time" example, that the equivalent of being offline 1 second in 10,000,000,000,000,000,000 years!

What does this mean?  It means that a homeopathis solution is effectively water.  Nothing more.  Yet, it is sold, and people buy it.  Homeopathy defies the laws of physics and chemistry, but due to effective marketing and a proper lack of FDA involvement, homeopathic "remedies" continue to be sold alongside legitimate medications.

One more reason to think twice before you consider purchasing Snake Oil.





Wednesday, February 3, 2010 ~ Have your say...

Lancet Retracts Controversial Autism Paper

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This is making news all over the place, and while I haven't had the chance to discuss the anti-vaccination movement in detail, this article is a great way to start:

Lancet Retracts Controversial Autism Paper



Monday, February 1, 2010 ~ Have your say...

Mars' Spirit Lonely Story... Poor Rover...

I wasn't sure whether to laugh or cry when I saw this cartoon:


For more info on our hero, visit NASA's website.  And to learn all about the Mars Spirit Rover, you can read the Wiki post.

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