The Doomsday Calculation

 

Consider answering the following questions:

How much longer will humans exist?

How long will a marriage last?

How much longer will that building be standing?

William Poundstone explains a simple approach to answering such questions in his book The Doomsday Calculation: How a Formula That Predicts the Future Is Transforming Everything We Know About Life and the Universe.

The Doomsday Calculation approach uses the Copernican principal which states that we are not in a special place (or time). Let’s take an example of you meet two couples for the first time at an event. The first couple has been together for a year, second couple for ten years. If you had no other knowledge of their relationships and you had to predict which couple would more likely be still together in a year, most people who probably pick the second couple. Your answer would be the same if you had to answer the question for being still together in five years or more. The first couple may be perfectly matched, but only given the evidence of the current length of their relationships, the second couple has demonstrated more longevity.

The Copernican principal states our observation of the couple is at no special time. In meeting the second couple who has been together ten years, we could be meeting them near the beginning of 70+ year marriage or we could be meeting them a year from a breakup, Let’s breakup the full (unknown) length of the relationship into quarters. We have a 25% chance of meeting them in the first quarter, a  50% chance of meeting them in the middle two quarters, and a 25% chance of meeting them in the final quarter.

The middle 50% is the range we’re interested in. If your meeting with them happens at the low end of this range, 25% of the length of their relationship has been ten years and they have another 75% or thirty years left. On the other hand, if ten years is the 75%, they only have another 25% left, which in this case is three years and four months remaining. With those two extremes evaluated, we can state we are 50% confident the relationship will last between three years and four months to thirty additional years.

Perhaps you want to state your range with a higher confidence level. That will mean widening the interval. Moving to 75% confidence, the interval becomes approximately a year and a half to seventy more additional years.

Poundstone gave a brief summary of the doomsday calculation with additional examples in: Doomsday argument: the end of humanity, predicted by a math equation - Vox

 

William Poundstone is the author of 16 books, including The Doomsday Calculation: How a Formula That Predicts the Future Is Transforming Everything We Know About Life and the Universe.

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Card Magic Based in Math

 

I recently viewed a magic card trick on YouTube demonstrated by MLT magic tricks. As with many card tricks, the technique is based on math. The trick is good because it doesn’t require “setting” the deck ahead of time. Also, the trick only requires minimal card manipulation and so it is a good trick for beginning magicians to master.

Rather than describing the set-up and execution of the trick, I recommend readers to the MLT YouTube video. Near the end of the video, he mentions that he knows the trick is based in math; however, he admits he doesn’t know the exact math behind the trick. Watch the video and return here to learn the math basis of the trick. Knowing the math doesn’t help in executing the trick but does allow one to variations of the trick.

Here's an abbreviated description of the trick execution:

Shuffle the deck (standard 52-card deck)
Allow the spectator to shuffle the deck
Split the deck into two equal piles of 26 cards each
Have the spectator choose one of the two piles
Have the spectator split the chosen pile into two smaller of stacks of any size
Have the spectator put aside one of the smaller stacks (we’ll call it the count stack)
The magician picks up the non-chosen pile and spreads out the 26 cards face down
The spectator picks one card from the fanned-out cards
The spectator places the card back on the pile
The magician shuffles the cards (with one minor manipulation
The shuffled pile is placed on top of the small pile of cards (the small pile not set aside)
The magician shuffles the combined pile
The spectator is asked how many cards are in the count stack
The magician counts off that number of cards and reveals the spectator’s card

Here is the math behind the trick:

The deck of cards has the standard 52 cards. When the deck is split, each of the two piles has 26 cards. The spectator choices one of the two piles and splits that pile into two smaller stacks of any size. The size of the stack set aside (we called it the count stack). The number of cards in the count stack will be designated c. Therefore, the number of cards in the stack not set aside, is 26-c.

When the spectator returns the chosen card to the magician, the manipulation forces the card to bottom of the 26-card pile. The magician places this pile on top of the remaining stack. The spectator’s card is in 26^th card from the top.

The final shuffle has the magician pulling the top and bottom cards of the combined stack in pairs (see video for visual description). The result of this final shuffle moves the spectator’s chosen card from the 26^th position to a position closer to the top by the same number of cards in the remaining stack. Earlier we determined this number to be 26-c. So what is the new position of the spectator’s card. We’ll call this position p. Initially, it was at 26. After the final shuffle it moves up 26-c places, so final position is:

p = 26-(26-c) =c

The position of the spectator’s card in the final deck configuration is the same number of cards in the count pile (c).

This blog has other posts related to math-based magic tricks:

Uncle Billy Magic Trick

Red/Black Magic Trick

Dice Magic Trick

Magic Trick with Card and Algebra

Phone Number Fun Based in e

Check out MLT Magic's Amazon page at: MLT Magic Trick's Amazon Page

The Frequency of Prime Numbers – The Prime Number Theorem

 

As we progress through the first twenty natural numbers, there are four prime numbers in the first ten numbers (2, 3, 5, 7) and another four in the next ten numbers (11, 13, 17, 19). So, in these first twenty, we have a relative frequency of 40% prime numbers. After this point, the percentage of prime numbers begins to drop. Under 40, there are 30% prime numbers; under 100, 25% prime. Moving to 1000, there are 168 prime numbers, so the frequency drops to 16.8%.

For centuries, this persistent drop of prime numbers has been known and is termed the Prime Number Theorem (PNT). It states that primes number distribution asymptotically drops. There is a function, called the prime-counting function and uses the expression, π(N), as the number of primes less than or equal to N.

A first approximation of this function is:

π(x) ~ x/ln(x)

A result of this function is that the nth prime number, p_n, can be approximated to:

p_n ~ n*ln(n)

The prime-counting function was also used in a prior post Math Vacation: Counting Prime Numbers (jamesmacmath.blogspot.com).