Fun with Numbers

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How fast monkeys can actually type Shakespeare

There is a radio ad playing right now that says that a monkey on a typewriter could type Shakespeare in 3 million years. Apparently the advertiser is aware of the old saying that an infinite number of monkeys on an infinite number of typewriters would eventually produce the complete works of Shakespeare.

But this radio ad talks of only one monkey, and gives how long it will take; 3 million years. That doesn't sound very long if the monkey is just randomly typing. Is it actually possible? Let's work it out.

If the monkey types at the speed of a good typist, then it is hitting about 150 words a minute, or about 900 characters per minute. That's about 50,000 characters per hour, or about 1 million characters per day, working 24 hour days. 3 years is about 1000 days, so 3 million years is about 1 billion days, or time to type about 1 million billion characters.

Now the complete works of Shakespeare is fairly small, say about a million words or so. That works out to about 10 million characters, so obviously the monkey has plenty of time.

But monkeys are not very good typists. They are fast, but they are not very accurate. So our 3 million year old monkey will have done a lot of typing, but most of it will have been errors. If we assume that the monkey only types letters (no numbers or punctuation), and we want the monkey to type a specific word, then the number of tries required will be about equal to the number of ways that many letters can be typed.

So, for a one letter word (like "a"), the monkey will probably get it within about 26 tries, or a couple of seconds. A two letter word, like "to", or "be", would take about 676 tries, or about 45 seconds. A three letter word, like "not" would take about 17,500 tries, or about 1200 seconds (which is about 20 minutes). The following chart shows how this grows.

Length of wordNumber of triesTime Example
1262 secondsI
267645 secondsto, be
317,00020 minutesnot
4450,0008.5 hoursthat
512 million9 dayssleep
6300 million240 daysnobler, suffer
78 billion17 yearswhether
8200 billion440 yearsquestion
95.5 thousand billion11,000 years perchance
10141 thousand billion300,000 years outrageous
113.6 million billion7.7 million years Shakespeare
1295 million billion200 million years undiscovered

So, could a monkey type Shakespeare in 3 million years? Well, if it was slightly faster than our monkey, then it might be able to, but only the word "Shakespeare".

Actually, statistically it takes about half the number of tries to get the result you want, so even our 3 million year old monkey could probably type "Shakespeare" in about 3 million years.

Would You Rather Pay No Tax, Or Get 15% Off?

Here in Ontario, the total sales tax is 15%, and people hate it. But retailers have come up with a way of fighting back. Instead of the old tactic of having sales offering 10, 15, or 20 percent off, they now have "pay no tax" events. This is great, because it's like you get the old 15% off, and you get to feel that you're beating the tax.

But wait. Is it really like getting 15% off? Let's take a closer look. If something costs $100, and the dealer pays the tax for you, then you walk out paying $100, which is what the sticker says. That's a pretty good deal.

But if the dealer gives you 15% off, then you have to pay him $85 ($100 - 15%), and then pay 15% tax on top of that. But note that you are now paying the tax on a smaller amount ($85), so the tax is now less than the $15 you would have paid. In fact, the tax is now only $12.75, so you pay $97.72, and walk out of the store with an extra $2.25 in your pocket.

What's going on here? The discount was 15%, and the tax was 15%, so they should cancel out, right? Wrong. When you get a discount and an increase of the same percentage, the result should always be less than you started with. If somebody tries to convince you otherwise, check their numbers!

Watch Out For Bargain Pricing

We've all been in stores where they have the same product in various size packages. I was recently in a store which had bars of soap in packages of 6 for $1.69 or 9 for $2.99. Which is the better deal?

I noticed that 9 is just an extra half of 6. So if I take the price of the 6 and add half of it again, then I find out how much 9 bars would cost at the price for the 6. $1.69 is about $1.70, and half of $1.70 is $0.85, so the total would be about $2.55 (give or take a couple of cents). But wait, the 'bargain' price for 9 bars is $2.99. It looks like it's not such a great bargain after all!

Sometimes the larger size really is a bargain. I remember one experience with potato chips where I could get 180 grams for $1.99, or the new 300 gram bag for $1.89. Even if I was only going to eat 180 grams, I was still better off buying the larger bag!

Fortunately, many stores now label their shelves with the 'unit prices', which are supposed to provide fair comparisons between different sizes and different products. It pays to check them out. If your store doesn't show unit prices, then carry a calculator, or do 'quick check' calculations to make sure you know which size really is the bargain.

(Psst - Want the real facts about unit prices? C'mon over here.)

When were Adam and Eve?

Has population always been growing at the present rate (about 3% per year)? Just for fun, let's assume that it has, and we can figure out when Adam and Eve must have started it all.

There are about 5 billion people now, and Adam and Eve were two people. So, the two things we need to know are how many times would the population have to double for Adam and Eve to become 5 billion people, and how fast does it double.

From 2 to 5 billion, we need to add 9 digits (the 9 zeroes in billion), and we need to double once to turn the 2 into a 5. Every ten doubles gives us 3 more digits, so we need to do that 3 times (30 doubles), plus one extra double, for a total of 31 doubles.

At 3%, the Rule Of 72 tells us we need about 24 years to double. So, if the population has always been growing like it is now, Adam and Eve must have been about 31 x 24 years ago. Let's see, that's about 750 years, or about the year 1200. If I remember my history correctly, Europe and Asia had flourishing civilizations by then, which makes it unlikely that Adam and Eve were just showing up.

Wait a minute... That doesn't make sense...

When Will the Earth Be Full of People?

Okay, so Adam and Eve were about 750 years ago, and since then we've been working to fill in the rest of the space on the planet. My friend Jeff Silverman asked me when we will finish. When will the Earth be covered with people? Let's see if we can work it out.

Right now, there are about 5 billion people on the planet. The radius of the Earth is about 6,000 km, or about 6,000,000 metres. That means that the surface area (given by 4 * pi * radius-squared) is about 4 * 3 * 6,000,000 * 6,000,000 square metres, or about 400 trillion square metres.

Of course, we can't have people everywhere; something like 70% of the Earth's surface is covered by water, which only leaves about 120 trillion square metres for the people. So, with 5 billion people on 120 trillion square metres, that gives us about 24 thousand square metres per person.

An acre is about 5000 square metres, so that's about 5 acres per person. Not exactly a lot of space, but not really crowded yet. If we say a family can live on an acre, that would mean we need 1 acre for 4 people, which is roughly 1/20 of what we have now. Based on the same doubling as we used for Adam and Eve, that would take about 4 doublings, or about 96 years. So, in about a hundred years, there'd only be about an acre per family.

But maybe that's still not crowded. After all, in the city, people live in apartments or houses which are only about 100 square metres. If we allow each of them to live alone, then that's about 1/240 of the area we each have now. It would take about 8 doubling periods, or about 200 years to get down to this little space. Since we'd still need work space and shops, that would be getting pretty crowded.

But there would still be room to move around. If everyone just had room to stand without getting crushed, we'd each need about a square metre (actually, you could probably get the family in there, but that might be stretching closeness a bit far). That would be about 1/24000 of what we've got now. 24000 is between 16000 (14 doublings) and 32000 (15 doublings), so let's be generous an give ourselves 15 doublings, or about 360 years before we can no longer walk around. Now that would be crowded!

- Tricks with numbers.
- Some guidelines for 'quick check' calculations.
- Some useful number facts for quick checks.
- Take the probability quiz.
- Wordlore and wordplay.
- Return to the front gate of Esmerel.

Page maintained by the Mathemagician.