Probability Question (Real-Life Example)

In My Humble Opinion
1

Two questions, actually.

Have the same exact numbers ever been drawn in a lottery? You realize that this is theoretically possible. And I don’t necessarily mean right afterwards. Maybe even decades later, for example. (And dopers from other countries, please chime in too :slightly_smiling_face: .)

And what is the smallest margin of popular vote that has ever won an election? I was originally going to ask if an election was ever swung by one vote. But I think that this question is more plausible.

Thank you in advance for your replies :slightly_smiling_face: .

2

Yes, this has happened a number of times.

3

I was elected president of National Honor Society my senior year in high school. I won by one vote beating out a good friend. We were both absent that day.

4

You got me beat. I lost the 4th grade election for class Secretary by 4 votes.

5

How many numbers to be drawn in the lottery? We have Pick 3 and Pick 4 games where the number range is 0-9.

6

We do, as well, here in Illinois – those are the Illinois Lottery’s “daily games.” With 1000 possible outcomes for Pick 3, and 10,000 outcomes for Pick 4, it’s almost certain that the same numbers have come up repeatedly in those games, over the years.

I suspect that the OP is thinking more of games like Mega Millions and Powerball, where there are millions of potential combinations. But, different lotteries use different number ranges, numbers of balls, and numbers of pools (e.g., Mega Millions and Powerball draw numbers from two different pools of numbers), making his question even more challenging.

7

And yet you still went on to make something of yourself! Le Chaim :clinking_glasses:

8

Worse yet, both those examples have increased the pool of balls over the years.

But, at any rate, a little Google-fu reveals that the same numbers did pop up in the Bulgarian lottery (6 numbers out of 49) twice in short order:

9

This is an example of the Birthday Problem, which states in its original version that if you have a room with only 23 people in it, there is a better than even chance that you’ll have two people in the room who share a birthday.

That’s for a “lottery” with 365 possible options, and 23 “draws” from the lottery. In general, to get a decent chance of a match, the number of draws needs to be somewhere in the vicinity of the square root of the number of possible outcomes in each draw (23 \approx \sqrt{365}) Thus, for a lottery with 100 million possible options, you’d only need somewhere in the vicinity of ten thousand draws to get a reasonable chance of a match. That’s about 30 years of draws, at one per day.

10

Sorry, I should’ve made myself more clear.

No, I’m thinking of much bigger draws. At least five random numbers, let’s say. :slightly_smiling_face:

11

Germany and the Netherlands had the same numbers (09, 17, 18, 20, 29, 40) drawn on two consecutive weeks in 1977.07.18:

You chose 6 numbers out of 49 and an extra number for the jackpot. 205 German punters chose those numbers, probably because they had been drawn the week before in the neighboring country, so that each pocketed a prize of only 30.737.80 DM, the second lowest prize money ever. The extra number for the jackpot was different though: 31 vs 42. By sheer probability one of the 205 should have guessed the correct 42, but the article does not mention it. If they all also chose 31, obviously, they did not.
ETA: That was before German Lotto used the additional number for jackpots, it was the time the additional number was used to draw the second highest prize, 5 number plus the additional one. The jackpot number is only one digit. So the numbers were identical for the highest prize. Which in this case was quite low, as stated.