You’re on a game show and the host asks you to pick one of three doors. Behind one of them is the star prize: a sports car. Behind the other two are goats. Once you have made your pick, the show host opens one of the other doors – always revealing a goat. The host then asks if you are happy with your original choice, or whether would you like to switch.
Would you switch? Think about it before reading on.
This problem was posed to the mathematical community in a letter to The American Statistician in 1975, but was popularised by Marilyn Vos Savant in 1991.
Marilyn’s article suggested that the right thing to do is to switch. If you read the article – and please do – you will see that Marilyn received many very negative comments – some would say rude, to her conclusion. Here are some examples:
“You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!“
“Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful.“
“Maybe women look at math problems differently than men.“
“You are the goat!“
It is incredulous that people could write such comments and – and it is now accepted that Marilyn was correct and the right thing to do is to switch doors. There would have ben some red faces and I hope they apologised.
If you are faced with three doors and choose one of those at random, I think we’d all agree that you have a one in three (1/3) chance of choosing the car.
If one of the other doors is opened, showing a goat, one argument says that switching doors now gives you a 1/2 chance of winning the car.
Another argument says that once you have chosen a door, you have a 1/3 chance of winning the car. If you could choose the other two doors instead you would have a 2/3 chance of winning the car. If you are now showed one of these doors – revealing a goat – by switching, you still have a 2/3 chance of winning the car. Of course, you will not always win a car but over many games, by switching you will win 2/3 of the time, compared with only winning a 1/3 of the time if you do not switch.
There are two ways that might persuade you that switching is the right thing to do.
Increase the number of doors to 50. When you choose a door you have a 1/50 chance of winning the car. This means that there is 49/50 chance that the car is behind one of the other doors. The host now opens all the remaining doors, just leaving your original door and one other. If you stick with your original door you still have a 1/50 chance of winning the car. If you switch, you have a 49/50 chance of winning the car.
We can run a simulation. In preparing this article, I wrote a Java program to do that. We ran the Monty Hall Problem 5,000 times, both switching doors and not switching. If we have three doors, by not switching, the car was won 1702 times, compared to 3310 when switching. These are close enough to the 1/3 and 2/3 win ratios that we would expect. If we now change the number of boxes to 50, if we don’t swap we win the car 99 times. If we swap, we win 4,914 times. Again, this is close enough to the theoretical figures to persuade us that switching is the correct thing to do.
Hopefully, you are now persuaded that you should switch doors when faced with a similar situation. Good luck!
Paul Erdös (1913-1996) is one of the most prolific mathmeticians. He wrote over 1500 papers in his lifetime and collaborated with over 500 people. As a tribute, his friends created the Erdös number, which is a tongue in cheek way of asking how well you are associated with the top mathematicians.
Erdös himself has an Erdös number of 0. A co-author of Erdös has an Erdös number of 1, if you have written a paper with one of those co-authors you have an Erdös number of 2, and so on.
I found out recently that my Erdös is 3. That’s not too bad. I can never get an Erdös number of 1, but perhaps (one day) I might write a paper with somebody who has an Erdös of 1, giving me anErdös number of 2.
If you are interested in finding out what your Erdös number is, I found two very good web sites:
Most people have heard of the mathematical constant Pi (?), and will know that it’s roughly 3.14. Taking inspiration from these three digits, March 14 (3/14 in the US date format) is heralded as international Pi Day, first marked by US physicist Larry Shaw in 1988.
This year brings a unique opportunity to demonstrate an entirely unnecessary degree of zeal by marking Pi Day correct to nine decimal places on March 14, 2015, at 9.26am 53sec – corresponding to 3.141592653, the first 10 digits of Pi. If you’re too busy this weekend, you could book in July 22 – another way of expressing Pi approximately is the fraction 22/7.
Pi is calculated as the ratio between a circle’s circumference to its diameter.
Pi is always the same value, no matter the size of the circle, which makes it an important mathematical constant.
The ancient Babylonians calculated Pi as three by taking three times the square of the circle’s radius, later refining the value to 3.125. Archimedes of Syracuse (287-212 BCE) approximated Pi by inscribing polygons on the inside and outside of a circle. By increasing the number of sides of the polygons, Pi could be calculated to higher levels of accuracy.
Even today, calculating Pi to ever increasing levels of accuracy continues – it has now been found to an accuracy of over 13 trillion digits. There’s no reason to suspect this record will remain forever, even though only about 39 decimal places are sufficient for astronomical precision. There is no reason to be more precise for practical purposes but it is good scientific sport of sorts to strive to be ever more accurate.
Some Properties of Pi
Pi is an irrational number, which means it cannot be accurately represented as a fraction, a/b, where a and b are integers. An approximation is to express it as 22/7 (3.1428…) which is inaccurate by 0.04025%. A closer approximation is 104348/33215, which has a far smaller error of 0.00000001056% but is still, technically, wrong.
Pi is also a transcendental number which, simplified, are numbers that cannot be reduced algebraically (more accurately a number that is not the root of any non-zero polynomial equation with rational coefficients).
The proof that Pi is transcendental was found in 1882, but it had been known for much longer that if Pi was transcendental then it would be impossible to square the circle – to construct a square with the same area as a circle.
Putting Pi to use
Among the unusual uses for Pi is its relation to the nature of meandering rivers. A river’s path is described by its sinuosity, it’s tendency to wind from side to side as it traverses a plain. This is described mathematically as the length of its winding path divided by the length of the river as the crow flies. The average river has a sinuosity of about 3.14.
Albert Einstein actually made some observations about why rivers meandered as they did. He noticed that the water that flows faster around the outside of a bend, eroding that bank more quickly. This creates a larger bend. These bends eventually meet and the river forms a short cut through them. Hans-Henrik Stolum used these observations and noted a relationship with chaos theory, which suggests that, despite rivers straightening out as the rivers cut through the short cuts, the sinuosity tends to move back towards Pi.
Further examples of where Pi appears in the real world can be seen in a BBC item written for Pi Day 2008, and this New Scientist item written for the 2010 Pi Day. For example, Pi can be found in the measurements of the Great Pyramid of Giza, the angular distances of stars in the sky and in a song by Kate Bush. Included in the lyrics were the first hundred digits, or so, of Pi, but she went slightly wrong at around digit 50.
How to celebrate Pi day?
If you fancy some Pi-related entertainment for Saturday, you could try:
Looking up whether your birth date appears in the decimal places of Pi – mine does, starting at 200,703, although if you want to know my age you’ll have to look it up.
Memorising the first digits of Pi. Piphilology, a system of mnemonics to help you remember the digits, may help. There are even piems (Pi poems) to help you remember. You may not beat the record-holder, which currently stands at 67,000 places.
Examining the first million digits of Pi – you might see a pattern no one else has.
Looking for Pi in everyday life. For example, it has featured in Mythbusters.
Albert Einstein, one of the greatest scientists the world has known, spent some time working on Pi as it related to rivers. Is it a coincidence that Albert Einstein was born on March 14, 1879? As he would have said himself, God doesn’t play dice.
I recently came across another blog (I’ll point you to that blog at a later date), which has a number of puzzles on it. One, in particular caught my eye (probably because of a long time fascination with conjuring).
The trick works as follows.
A volunteer chooses five cards from a normal pack of cards. You take the five cards, look at them and return one of the cards to the volunteer. The other four cards are handed to another volunteer. You leave the room and a colleague enters. The second volunteer reads out the cards you gave them and your colleague names the card that the first volunteer is holding.
How is this done (post your ideas)?
If you want to know, click the “interesting” box at the bottom of this blog and, if there is enough interest, I’ll point you to the original blog and the answer (unless somebody already knows, of course).