The Monty Hall Problem

the Pledge

Imagine you’re in a game-show. You have managed to blaze through the quarter-finals and semi-finals and end up as one of the two finalist contestants at the threshold of receiving the Grand Prize of 1 Million Dollars!

MHP all closed

Your final challenge is a simple guessing game. There are three doors: one of which contains the Grand Prize, and the remaining two contain a beautifully groomed and domesticated goat each. You must guess which door contains the Grand Prize. If you guess correctly, you receive the Grand Prize and are declared official winner of the game show. If you guess incorrectly, you will receive the goats, leaving the Grand Prize to your opponent.

MHP 3 open GMG

You have no way of knowing which door conceals which item. Whichever door you pick, you will receive the prize behind it.

The game-show host asks you to pick a door. Let us say you had a good feeling about Door A and picked it. Here’s the twist. Before the game-show host reveals the contents behind Door A, he reveals to you the content of one of the remaining two doors. The game-show host will always reveal the location of one of the goats. Let’s say he opened Door B, which contained a goat.

MHP AGC

Then game-show host offers you a second chance: do you want to stick with your initial choice, or would like to switch doors?

MHP A'Host'C

The question is:

Should you swap?

Should you stick with your original choice?

Or does it really make a difference?

the Turn

If your answer is  that it doesn’t really make a difference whether you switch doors or not, then you are unfortunately, very very wrong. Surprising isn’t it? It in fact does make a difference, a very large one in fact, as you actually double your chances by switching from your initial choice. Don’t believe me?

Never fear, Mathematics is here! 

the Prestige

Assuming you are among the many who believed it didn’t make a difference (if you’re a smarty-pants and chose correctly to swap doors, then go be smart some place else…), then you probably made the false assumption that there was a 50% chance that the Grand Prize would be in Door A as well as Door C.

MHP SG-G-SM MHP SM-G-SG

Here’s an example.

We will use the same arrangement, where Door A contains a Goat, Door B contains a Goat, and Door C contains the Grand Prize.

MHP G-G-M

Let us set some cases:

1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat]

MHP PSG-G-SM

2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat]

MHP G-PSG-SM

3. You pick Door C [it contains the Grand Prize], the Host shows you Door A/B [it contains a Goat]

MHP SG-G-PSM

Let’s count the number of times you win if you do not switch doors and remain with your initial choice.

1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat], you remain with Door A.

MHP PG-G-SM

YOU LOSE

2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat], you remain with Door B.

MHP G-PG-SM

YOU LOSE

3. You pick Door C [it contains the Grand Prize], the Host shows you Door A/B [it contains a Goat], you remain with Door C.

MHP SG-G-PM

YOU WIN

Therefore:

  • P(Lose if you do not switch) = 2/3
  • P(Win if you do not switch) = 1/3

Let’s count the number of times you win if you switch doors rather than remaining with your initial choice.

1. You pick Door A [it contains a Goat], the Host shows you Door B [it contains a Goat], you switch to  Door C.

MHP PSG-G-PM

YOU WIN

2. You pick Door B [it contains a Goat], the Host shows you Door A [it contains a Goat], you switch to  Door C.

MHP G-PSG-PM

YOU WIN

3. You pick Door C [it contains the Grand Prize], the Host shows you Door B [it contains a Goat], you switch to  Door A.

MHP PG-G-PSM

YOU LOSE

Therefore:

  • P(Lose if you switch) = 1/3
  • P(Win if you switch) = 2/3

You can also come up with the same answer by drawing a tree diagram,

MHP Tree Diagram

or using Conditional Probability calculations,

MHP CP

As you can see, the Probability of Winning doubles when you decide to switch doors and the Probability of Losing halves.

This problem is known as the Monty Hall Problem or Monty Hall Paradox, and is named after the host of the TV Game Show “Let’s Make a Deal”. This problem is recognized for its counter-intuitive result – a Veridical Paradox.

For many years, it has baffled people, especially renowned mathematicians, for its paradoxical solution. Numerous simulations have been created to test whether you are more likely to win if you switch doors. The New York Times created a Flash Simulation on their website that allows you to test for yourself the Monty Hall Problem.

The Mythbusters and many other science shows have all tried their hand at this problem, and switching has always proven to be the most profitable choice.

The most important lesson to take from the Monty Hall Problem is to realize that our gut/instinctive feeling is not always correct, and that we must always be skeptical of everything.

 

Works Cited

1 – 2 – 3 – 4 – 5 – 6

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Contact – Probable Impossibility or Improbable Possibility?

Introduction

It was a dry and blistering August 15, 1977 at the Ohioan Big Ear Telescope. SETI (Search for Extra-Terrestrial Intelligence) Radio Astronomer Jerry Ehman spent this afternoon laboriously examining the shelves of printed perforated paper that detailed radio signals emanating from the cosmos. Hours spent sc800px-Wow_signalrutinizing over every seemingly inconsequential signal led to him to come across an anomaly curiously inconsistent with the library of signal data. The 6EQUJ5 code circled with his red pen indicated a radio signal thirty times stronger than any other normal signal received and a frequency ominously identical to the hydrogen atom’s resonance frequency. Ehmer’s comment of the code – Wow! –  modestly sums up the discovery. Was it first contact? Was it just an emission from celestial bodies like quasars, known for their violent ejections of electromagnetic energy? Alternatively, was it an alien races’ misguided attempt at invading our planet by kidnapping the Earth’s then “King”, Elvis Presley, who coincidentally “passed away” the following day?

Wow_signal_profile.svg

This 72 second supposed alien message has since chosen not to reveal itself after our scouring the skies for a signal with the slightest resemblance. Whatever the origin and cause, the “Wow! Signal” had become an iconic symbol for the SETI’s goal, and by extension, humanity’s evolution as a technological and cosmological species. The implications of such a discovery were subject to our imagination. As Carl Sagan, astronomer once said, “Imagination will often carry us to worlds that never were. But without it we go nowhere”.

The Drake Equation

An example of this use of imagination is the Drake Equation.

11drake-equation

The Drake Equation,  a mathematical equation created by astronomer Frank Drake in 1961, is used to approximate the number of detectable intelligent extraterrestrial civilizations in the Milky Way, and by extension our chances of encountering them. As you can see, the equation involves a variety of variables:

  • N = the number of civilizations in our galaxy with which communication might be possible
  • R* = number of new stars born each year
  • fp = percentage/fraction of stars with planets
  • ne = average number of habitable planets per solar system
  • fℓ = percentage chance a habitable planet develops life
  • fi = percentage chance that life develops intelligence
  • fc = percentage chance life can communicate across space
  • L = the length of time for which such civilizations release detectable signals into space

If you notice, the type of information for each following variable becomes more and more specific; it calculates the approximation from a macro-perspective and narrows the specifications of the variables. Essentially, the equation is structured using the Multiplicative/Fundamental Counting Principle.

The chances of finding intelligent extraterrestrial life in our galaxy can either be extremely high or extremely low. We must understand that the Drake Equation presently exists to provide perspective, rather than calculate a specific number. With the level of technology and knowledge we have so far, we can accurately come up with values for the first three variables by observing the billions of galaxies and stars in the night sky. But the values for the remaining four variables is up to our own pessimism or optimism and are the ones scientists are searching for. The problem with trying to predict the chance that a planet develops life that becomes intelligent that achieves interstellar communication is that we have only one data point, and that is us on our pale blue dot.

However, the Drake Equation is not necessarily perfect, as there have been numerous refinements and abridgements to it. For example, the nr variable describes the percentage chance of new civilizations evolving from previously extinct civilizations. This is an optional variable for the equation as it takes into consideration the billion year lifespans most terrestrial planets possess.

Each variable is laden with assumptions; for example, the fp variable factors only the number of terrestrial planets rather than gas giants Saturn and Jupiter. This makes the assumption that gas giants are unable to host intelligent life, as far as we know.

If we look at the types of values that would be used in the estimate, the values R*, fp, and ne are closer to powers of 10 rather than 1. The fl, fi, fc factors would all be numbers less than 1 (

10^-1 or even smaller). If we multiply these values together we can approximately get a value of 1. Therefore, based on these crude estimates, the number of civilization in our galaxy with which communication might be possible, N, is approximately equal to L, the longevity of a civilization.

n=l

Probabilities of there being intelligent aliens, when it comes to something as infinite as outer space, has the potential of being extremely large. Because space is so large and the formation of planets and other phenomenon is seemingly random, there is undoubtedly at least one (apart from our own) planet with intelligent life. This is analogous to finding one’s phone number in the digits of pi or any other transcendental number like e or phi. Because of pi’s random digits, there exists a chance of finding any conceivable number permutation, based on the notion of infinity and randomness. Based on these assumptions, that the universe is in fact infinite (or just terribly large) and its processes are random in nature, intelligent life should exist elsewhere. The issue is whether we can contact them or not.

“Where are they?”

– was the big question asked by Italian Physicist Enrico Fermi. If Earth is considered an atypical planet orbiting an ordinary young yellow star like billions of other planets, there should be a multitude of civilizations in space – yet where are they? So far, we have not one strong piece of evidence supporting this assumption, just evidence of the contrary.

This idea is called the Fermi Paradox – “the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilization and humanity’s lack of contact with, or evidence for, such civilizations”.

There are numerous reasons for this being the case:

  • The Universe is unimaginably large – the diameter of the observable universe (notice the word “observable”) is 93 billion light years. Intelligent life can possibly exist anywhere.
  • The Speed of Light is finite – the speed of light in a vacuum is approximately 299 792 458 m / s. It would take a photon of light 93 000 000 000 years to travel from one edge of the observable universe to the opposite end.
  • The Universe is still expanding – as discovered by astronomer Edwin Hubble from observing Doppler shifts from neighbouring celestial objects, the Universe is still expanding after 13.77 billion years. Like ants on an inflatable balloon, they get farther and farther apart even if they stay still. Similarly, everything is essentially flying away from us, and shouting “come back” into the night sky won’t help.
  • Radio Transmissions never get to reach us – radio waves are just low frequency electromagnetic radiation or light. Light, in the presence of a large mass like a star or black hole, will have its path bent due to gravitational influence. Because of this, not only do the chances of alien radio transmissions reaching us slim, but our ability to accurately locate its source will reduce.

There are other explanations for this paradox:

  • We are in fact alone – perhaps it takes extremely precise conditions for intelligent life to arise. This is known as the Rare Earth Hypothesis. This goes against the Mediocrity Principle – that Earth is like any other planet in the universe and thus the development of intelligent life is common. Proof against the Rare Earth Principle is the thousands of exo-planets that we have discovered that have Earth-like conditions.
  • We’re not looking properly – perhaps aliens are transmitting data on frequencies unknown to us, or have encrypted transmissions that we might pass for insignificant.
  • Intelligent Life tends to destroy itself – perhaps developing as an intelligent species inadvertently killed them in the process such as by destroying the ecosystem or mutual assured destruction through war.
  • Natural disasters eradicate Intelligent Species – perhaps earthquakes, volcanic eruptions, emerging viruses, or meteor impacts may have caused intelligent life to die out.
  • We have not searched long enough to detect and comprehend interstellar transmissions

There are also other explanations that can be slightly 1984-esque:

  • We are purposely not contacted – perhaps extraterrestrial intelligence has developed and has decided that we never contact them. This is known as the Zoo Hypothesis, where we would be the animals in the cage being observed. Their reasons for doing so? They might be conducting experiments on us. They might be waiting for us to reach a higher technological level before contacting us.
  • The Alien Civilization does not agree with itself. The question of who speaks for Earth/Humanity might be similar to what other civilizations might undergo.
  • It would be dangerous to communicate with other alien civilizations – perhaps alien civilizations have found out that contacting other civilizations might be disastrous, and so have decided to avoid us.
  • They are here on Earth unobserved – We might underestimate their size or form. They could be microscopic or they have taken our own human forms. Perhaps they are here among us. Maybe its the bus driver, your neighbour, your teacher…

Conclusion

Calculating the probability of  intelligent extraterrestrial life in the Universe is subject to numerous variables. Our chances of encountering them is largely determined by our physical constraints and perhaps extraterrestrials’ decisions

As Arthur C. Clarke once said: “Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.”

Works Cited

1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10

Deriving the Power Rule using Binomial Theorem

This derivation goes to show once again the pervasiveness of mathematical concepts in different areas of the field. While Euler’s Formula does a better job of doing that, this simple derivation will suffice.

The Power Rule is often the first Derivative Rule many Calculus students will learn. It is commonly used to calculate the derivatives of polynomial functions, as well as reciprocal functions and nth roots of functions.

There are numerous and varied derivations and proofs on the internet that show the derivation of the Power Rule, but the one I will focus on involves using the Binomial Theorem through the First Principle or Difference Quotient.

Power Rule Binomial Theorem 3

When trying to come up with this derivation, I tried to avoid expanding the Summation. But evaluating the limit alongside manipulating the summation made it harder to see a clear set of steps to derive the Power Rule.

This proof makes you wonder (at least me) how Isaac Newton (or Gottfried Liebniz, but nobody really cares about him…) derived this Derivative Rule. Was it through observing the patterns in different polynomials? Was it using this exact derivation? Or was it…

aliens

Interestingly, the Power Rule was actually created by Liebniz rather than Newton, as commonly believed. Newton’s method of calculating polynomial derivatives was more complicated, thus being the reason why we continue using Liebniz’ notation. Newton and Liebniz were both contemporaries in the 17th century, with Newton being English and Liebniz being German. Newton had made his discoveries in 1666 and had them published in 1693, while Liebniz had made his in 1676 and published in 1684, sparking much controversy. This controversy was taken to court as it had become a matter of national pride for both countries. Being credited as the country for first inventing Calculus – then considered the forefront of mathematical innovation – would be immensely advantageous. As a result, it was taken to court under the mediation of the Royal Society (a society for the sciences – still present today). Coincidentally, Isaac Newton was the society’s president, thus leaving poor Gottfried Liebniz with charges of plagiarism. This story reveals the false perceptions we have of those in history.

newton_and_leibniz

Who would have considered Newton as someone who abused their own authority? It is similar to wondering if Leonardo Da Vinci was a bigot or Ludwig van Beethoven a drunkard.

If it is any consolation, England was far behind Europe for most of the 18th century in terms of mathematical innovation due to Newton’s complex notation.

In conclusion (a terrible way to end, but it gets the job done nonetheless), this specific derivation of the Power Rule reveals the interconnectedness of mathematical fields. The Binomial Theorem is useful in numerous situations and contexts, and this derivation proves so.

Works Cited

1 – 2 – 3