Six Numbers for My Grandfather
When I was a child, my grandfather played the lottery often. The game asked him to choose six numbers from 1 to 49, and sometimes he asked me to pick them.
I usually answered with 1, 2, 3, 4, 5, 6, or perhaps
2, 4, 6, 8, 10, 12. I was teasing him. He knew it too. He would object, ask me to
be serious, and refuse to waste money on my unserious guesses. To both of us, those numbers
looked like jokes rather than possible lottery results.
My grandfather also believed that people were bad at choosing random numbers. He was right about that. We tend to avoid repetitions, spread numbers too evenly, and mistake a lack of visible patterns for randomness. I explored some of those habits in an earlier post about randomness.
His solution was wonderfully practical. He made a small lottery of his own, drawing numbers from a bag as if he were playing bingo. The bag would choose without human bias. Surely that had to produce a better ticket.
How Many Tickets Are Possible?
Years later, after studying statistics, mathematics, and physics, I saw the puzzle differently. A 6-from-49 lottery has
\[ \binom{49}{6} = \frac{49!}{6!43!} = 13{,}983{,}816 \]possible tickets. Order does not matter, and a number cannot appear twice. A fair lottery gives every one of those tickets the same probability.
Each ticket above has a 1 in 13,983,816 chance. My joke was a perfectly valid guess. Grandpa's bag could choose a more random-looking ticket, but the lottery machine gave that ticket the same chance as mine.
What Are We Actually Comparing?
This is where semantics starts to matter. What does 1, 2, 3, 4, 5, 6 represent in
the comparison? It can mean one exact ticket. It can also stand for a broader idea such as
"a recognizable sequence." Those are different events with different probabilities.
Compared as exact tickets, 1, 2, 3, 4, 5, 6 and
2, 8, 11, 16, 42, 48 are equally likely. Compared as representatives of two
families, recognizable sequences are much rarer than combinations without a recognizable sequence.
Our intuition quietly moves from the first comparison to the second. That small change in meaning produces a completely different probability question.
Counting Consecutive Tickets
There is only one ticket equal to 1, 2, 3, 4, 5, 6. There are 44 tickets made from
six consecutive numbers. The first begins with 1, the next begins with 2, and the last is
44, 45, 46, 47, 48, 49.
The probability that the winning numbers form any consecutive run is
\[ P(\text{six consecutive}) = \frac{44}{\binom{49}{6}} = \frac{1}{317{,}814}. \]That is about 0.000315%. The probability that the winning numbers are not consecutive is about 99.999685%.
The second joke ticket, 2, 4, 6, 8, 10, 12, belongs to the family of arithmetic
progressions. In a 6-from-49 lottery, there are 216 six-term arithmetic progressions. The chance
of drawing one is about 1 in 64,740.
| Description | Compatible tickets | Probability |
|---|---|---|
| One exact ticket | 1 | 1 in 13,983,816 |
| Any six consecutive numbers | 44 | 1 in 317,814 |
| Any six-term arithmetic progression | 216 | About 1 in 64,740 |
| Any non-consecutive ticket | 13,983,772 | About 99.999685% |
So the sentence "1, 2, 3, 4, 5, 6 is less likely" can be true or false depending
on what it means. It is false when we compare that exact ticket with another exact ticket. It is
true when the sequence stands for a small, precisely defined family and we compare that family
with its much larger complement.
Microstates, Macrostates, and Entropy
Physics gives us useful language for this counting problem.
A microstate is one exact ticket. 1, 2, 3, 4, 5, 6 is one
microstate. So is 2, 8, 11, 16, 42, 48.
A macrostate groups many microstates under one description. "Six consecutive numbers" is a macrostate containing 44 tickets. "Numbers that are not consecutive" is a much larger macrostate containing 13,983,772 tickets.
Boltzmann's entropy counts how many microstates fit a macrostate:
\[ S = k_B \ln W. \]The quantity $$W$$ is the number of compatible microstates. Larger $$W$$ means larger entropy. The non-consecutive macrostate has far greater entropy because almost every possible ticket belongs to it.
This is close to what our eyes do when they see lottery numbers. We recognize
1, 2, 3, 4, 5, 6 as a member of a tiny, easy-to-name family. A messy-looking result
gets dropped into a huge catch-all family. We compare the families even when the original
question asked us to compare two individual tickets.
What Counts as a Recognizable Sequence?
The phrase "recognizable sequence" has no single mathematical definition. It might mean six consecutive numbers, any six-term arithmetic progression, only odd numbers, shared final digits, or a shape on the playslip. Someone else may notice a pattern that means nothing to me. These categories can also overlap.
We only get a probability after choosing a rule and counting how many tickets satisfy it. Six consecutive numbers gives us 44 microstates. Six-term arithmetic progressions gives us 216. A broader definition of recognizable patterns would give us more. The semantic choice determines the macrostate, and the macrostate determines the count.
Explore the macrostates
Choose a description to see how its number of compatible tickets changes the probability.
Bar lengths use a logarithmic scale. Arithmetic progressions include consecutive runs, so the rows are separate comparisons rather than parts of one partition.
Information Gives Us Another View
Shannon measured the information in an event as
\[ I(x) = -\log_2 P(x). \]Under a fair lottery model, each exact ticket carries about 23.74 bits of information because each has the same probability. Learning that the result was six consecutive numbers carries about 18.28 bits because that event happens only once in 317,814 draws. Learning that the result was non-consecutive tells us almost nothing because nearly every draw has that property.
Patterns also give us shorter descriptions. "The first six positive integers" is an easy way
to communicate 1, 2, 3, 4, 5, 6. An irregular ticket may be easiest to communicate
by listing every number. This idea is often called description length or algorithmic complexity.
Short descriptions feel planned. Long, awkward descriptions feel random.
Entropy counts the size of a family, Shannon information measures surprise under a probability
model, and description length captures how simply we can state an outcome. The meaning we assign
to 1, 2, 3, 4, 5, 6 decides which probability question we are asking. That is why the
same sequence can be an ordinary individual ticket and an unusual kind of result at the same time.
What Did the Bag Improve?
The bag solved the problem my grandfather wanted to solve. It removed his habits from the selection process and produced combinations without a human preference for certain numbers or shapes.
For the actual jackpot, the lottery machine only cared about the ticket he held. Once he had one valid combination, there was exactly one winning draw that matched it among 13,983,816 possibilities. His personal method for arriving at that ticket did not change that count.
That is the part I missed as a child. My sequence looked like a bad attempt at randomness because it was easy to describe and belonged to a tiny family of obvious patterns. It remained one valid ticket among millions.
The Question I Never Got to Ask
I still wonder what would have happened if my grandfather's homemade lottery had dropped
1, 2, 3, 4, 5, 6 onto the table.
Would he have played them?
Probably not but I guess we will never know.