# Why can't I teach that picking (un)popular integers doesn't lower Probability of winning lotteries?

Every year at my secondary school, some student always insists that "picking unpopular numbers" lowers your probability of winning lotteries, as follows.

Presuppose that lotteries let players pick any integer up till N≥32. Winning numbers range randomly from 1 to N. But you artificially narrow yourself to [32,N]. By disregarding [1,31], you flout the lottery’s random distribution of winning integers! As [1,N] contains more integers than [32,N], picking numbers ∈ [1,N] proffers more chances to win than picking ∈ [32,N]. QED.

First, I show them the quotes below†. Then I try to gainsay their flawed reasoning as follows.

By disregarding [1,31], you flout the lottery’s random distribution of winning integers!

This is a non sequitur. Popularity is irrelevant. All numbers have equal chance of winning. No choice changes your odds. "The only thing you can do to improve your situation is to pick an unpopular set of numbers to reduce the number of ways you split the prize should you win."

As [1,N] contains more integers than [32,N], picking numbers ∈ [1,N] proffers more chances to win than picking ∈ [32,N]. Q.E.D.

Picking any set of random numbers ≤ 31 gives the exact same odds of winning as picking any set of random numbers ∈ [32, N]. If the draw is random, every possible choice has the same odds as any other.

But I never convince them! They repeat this mistake on the exam! How can I correct their reasoning? How can I improve their intuition?

Here’s something you can do that might have an impact on your game: studies show that most people play numbers based on special days of the month, such as birthdays and anniversaries. So, numbers greater than the number of days in the month, or over 31, are less frequently played.
If you do win with a less popular number, there may be less people to split a prize. So while picking these numbers won’t increase the odds of a win, it might increase the amount of a win.

Anyhow, if you do play the lottery, you should not play obvious combinations, or anything that can be described in one sentence, such as "the birthdays of my pets, increased by 5" (yes, there is going to be someone who has pets born on the same days as you, and who also thinks 5 is his lucky number).

You should also not pick the date - or any date for that matter.

• Could one aspect of the fallacy be that by focussing on the range of possible values for the selection, it's easy to forget that the end product is only one value?  The aim is not to pick a range with the biggest chance of containing the lottery result, but maybe that's what's going on at the back of the mind.  — To replace that, it might be worth considering that by selecting a value in a restricted range, you're increasing your chance of winning if the result is in that range, even while reducing (to zero) your chance if it's not. Commented Jun 7 at 0:31
• @gidds yes! want to elaborate in an answer? Commented Jun 8 at 20:56

## 3 Answers

I'd say (time permitting) put it to a practical test.

Take a standard 6-sided die, say you're going to roll it 100 times. You (the instructor) guess "1" every time, while the student can pick whatever number they like. See if one or the other have a significantly greater number of wins.

If programming skill is available, write a (Monte Carlo) simulator that runs the same experiment on a larger scale (and/or with a wider range of permitted numbers). You write one function that guesses "1" every time; the student can write a function of whatever guessing algorithm they like. On a large run of trials (say, N = 100,000 or more), see if one or the other has a significantly greater number of wins.

My experience is that sometimes the act of formalizing the algorithm via programming causes the light bulb to go off and the querent to realize their mistake (e.g., seen it happen for the Monty Hall problem).

Another take (possibly if time for above is lacking): Simplifying the problem to the six-sided die as above, consider strategy A as "guess 1", and strategy B as "guess any random number from 1 to 6". Challenge the student to compute precisely how much more likely strategy B is to win. (Show work, of course.)

Did you consider the fact that your students have a point?
When playing the lottery, let's say that the numbers $$1$$, $$2$$ and $$3$$ are popular, while $$4$$ is not.
Being a popular number means that multiple people are guessing those ones, so let's say that four people vote for $$1$$, three vote for $$2$$ for $$3$$, while nobody votes for $$4$$.

Let's say that the price you get for voting the right number equals one thousand coins.
When you vote $$1$$ and you win, it gets split between you and the four other people and you get two-hundred coins.
When you vote $$2$$ or $$3$$ and you win, it gets split between you and the three other people and you get two-hundred and fifty coins.

... but when you vote $$4$$ and you win, you don't need to split and you get the whole nine yards!

The probability of winning stays the same ($$25\%$$), but the expected output is different.

• I believe this is exactly what OP is saying, and their students are responding that that "limits your choices". Commented Jun 3 at 12:01
• This aspect is already addressed in the OP's linked quote, "... So while picking these numbers won’t increase the odds of a win, it might increase the amount of a win." The OP's question is about the first clause there; nothing in the reputed student response is talking about prize-splitting. Commented Jun 3 at 13:03
• @DanielR.Collins Thanks for answering for me. You are dead right. with respect, this answer doesn't answer my question at all. Commented Jun 8 at 20:54

This is probably more of a psychological point than a mathematical one, but maybe it's easy to focus too much on the idea of having a range of values to choose from, and lose sight (at least subconsciously) of that fact that at the end of the process, what you're actually choosing is a single value, not a range.

If you were picking a range, then of course you'd want to pick the one with the highest possible chance of containing the lottery result — which is naturally the entire set of possible values (e.g. [1, 50] for some lotteries) — and in that case, restricting to a sub-range such as [1, 31] would indeed reduce the chance of winning.  So I wonder if that may be what's going on in the back of some students' minds?

So it may be worth stressing that that's not the case; that you're choosing a single value, and that any range you may consider along the way is simply a way of making that single choice, and evaporates once it has done its job and that choice is made.

If it helps, you could also point out that by choosing a value from a restricted range, you're increasing your chance of winning if the lottery result is in your range even though you're also reducing (to zero) your chance if it's not.  (In the extreme case, choosing from the range [1] gives a certainty of winning if that number wins, just as it gives an impossibility of winning if any other number wins.)

And those two effects cancel out (which you can go on to show mathematically — perhaps a useful exercise), so the overall chance of winning is unaffected.

(And, as your links point out, although choosing an unpopular value won't affect your chance of winning, it will reduce your chance of having to share the winnings, and so increase your expected winnings overall.  So it does make sense to try to discover the popular numbers, so you can avoid them!)