What are the odds? (Lottery question)

In My Humble Opinion
1

This is not a question about winning the lottery, so please read the OP before answering.

I play a multi-state lottery game called Lucky For Life. You choose five numbers from 1 to 48, then one Lucky Ball number from 1 to 18.

Each week i buy a ticket with five quick picks on it, where the lottery computer randomly selects my numbers.

My question is about the Lucky Ball numbers. If i have five Lucky Ball numbers chosen at random from 1 to 18, what are the odds that the same number will be drawn twice? I have noticed that almost every time i buy my ticket, one number is drawn twice. Not always the same number of course, but two of the Lucky Ball numbers will be repeated. Last night, three Lucky Ball numbers out of five were the same.

I’m no math wizard, so your help is appreciated.

2

This looks to me like a variation of the birthday paradox. Plugging in the numbers in this caculator, I get about a 46% chance at least 2 match.

Wolfram Alpha gives me the same answer, with 1 in 35 that at least three match.

3

It’s simple to work out the probability that all the picks are different:

After your first pick, there remain 17 unpicked numbers, so the probability that the second pick is different is 17/18.

If the first two picks are different, there remain 16 unpicked numbers, so the probability the the third pick is also different is 16/18.

etc.

So the probability that 5 picks are different is
(17/18) * (16/18) * (15/18) * (14/18) = 0.54

What you’re looking for, the probability that all picks are NOT different, is the complement of that:
1 - 0.54 = 0.46

(Note that this is the probability that AT LEAST two are the same, it’s a little more involved to work out the probability that EXACTLY two are the same, but the anwer won’t be too much different, since more than two the same is much more unlikely.)

4

Thanks, that sounds about like my experience. It just seems like i get three in a row that have pairs, and it feels like it happens more often than 46% of the time.

5

Don’t read too much into what happened in the past. Consider it this way: there are a lot of people playing, so statistically a few people will always get unusual runs like this… and the people that do will be the ones who might be motivated to ask - “that’s strange, I wonder what the probability is?”

If it keeps happening in the future significantly more than 46% of the time over a significant period, then there might be grounds to start suspecting that it isn’t selecting all numbers with equal probability.