At 1st September 2020, the number "999997" was picked for the first prize in Thailand's government lotto. The consecutive repeating of the number "9" caused extensive controversial discussion whether the lotto machine was working properly or not, some even claim that this incident proved that the government was cheating.
Note for the lotto drawing method. A six-digit number will be randomly picked from the set of 000000, …, 999999 for the 1st prize by using 6 staffs each draw a number 0 – 9 from their corresponding machines.
To simplify the problem, I will consider the 1st prize number "999999" instead of "999997" in this question.
Commonly, most people know that every number has equal probability of $1/1000000$. Let me define the mathematical statement for this.
Statement 1: Randomly drawing a number $n$ from the set of six-digit numbers $000000, …, 999999$, the probability of $n$ being any specific number in the set is $1/1000000$
Now, the problem arises when someone proposes the following statement.
Statement 2: Let $A$ be a set {000000, 111111, 222222, …, 999999}, The probability of $n$ being a member of $A$ is $10/1000000$.
On one side, people use Statement 1 to explain that the number "999999" being drawn is as usual as any familiar number such as "326648", "863439", …
On the other side, people use Statement 2 to claim that the number "999999" being drawn is "unusual" as it has only $10/1000000$ probability to draw this kind of number.
I got some feeling that latter claim using Statement 2 has something wrong because if I let the set $A$ being a set of my any desired 10 numbers such as {123456, 443253, 857342, …}, I could claim that any number is unusual. But I cannot explain it clearly enough to convince the people who believe this claim.
Please help me see if there is some mathematical explanation behind this conflict, which can explain why the claim using Statement 2 is invalid and why people find it difficult to figure it out spontaneously.
Best Answer
As said in comments, since there isn't anything considered "unusual," it's hard to define. Your logic is completely right. If it was $345678$ for example, there would be a similar story.
We think that a number like $999999$ would come up very rarely, but it comes up just as much as any other number, as you said. But for your statement 2, it goes with any other set $A$, for example $0000001, 111112, \dots, 999990$. But your statement is completely correct.