I am having an argument with my girlfriend. I live in the UK and our lottery has 48 numbers, of which 6 are drawn, if you match all 6 you win the top prize. I am arguing that I have exactly the same chance of winning if I choose the numbers 1,2,3,4,5,6 as any other combination of numbers chosen to play. Please help me here, as she has said that she will refuse to believe that I have exactly the same chance to win as her spaced out choice of numbers. She’s laughing in the background as I type this.
Of course, any set of numbers is as likely as any other, as you already know. And it’s only very slightly less likely than my winning, and I don’t buy lottery tickets!
However, if you choose numbers that others are less likely to also choose, there’s less of a chance that you’ll have to split the pot with someone else if you do win.
Slightly MORE likely I should have said.
Tell your girlfriend to stop laughing, SideshowBob. All combinations of numbers are equally likely, even if they’re in sequence. (Assuming, of course, that the lottery is fair.)
Do they draw with or without replacement? I’m not sure how lotteries work.
Either way, 1,2,3,4,5,6 has the same chance as any other possible outcome.
Hmm, I’m no expert in probabilities but my take:
There is an equal chance of any single number being picked (in this case 1 in 48).
But when you start talking about a sequence of numbers, the probability of any one sequence coming up begins to take a nosedive the more numbers you start to add.
So is 1,2,3,4,5,6 more likely to come up than any other combination?
Nope, as long as the drawing is truly random, that particular sequence of numbers is as likely to come up as any other sequence of numbers.
I know that 1,2,3,4,5,6 are just as likely to win, I know this because of the laws of probability. However she just can’t see it, she claims that getting 6 numbers in a row is less likely. She wants to know if you would play 6 consecutive numbers in all consciousness that you stand the same chance of winning. She firmly believes that it’s different having a sequence, I know this isn’t true and that any 6 numbers are just as likely, but will she listen.
BTW to former poster, no they don’t replace the balls, so no number can be drawn twice, it will always be 6 distinctive numbers.
Perhaps you can explain it to your girlfriend as such:
Imagine instead of being numbered in order, each ball had a nonsequential symbol of some sort on it. Say, one ball has a star, another has a clover, another a smiley face, etc.
Now since there are no sequences to be made, it should be clear that drawing “heart, smiley face, star, red stripe, American flag, green dot” is exactly as likely as drawing “clover, Nike swoosh, yinyang symbol, cross, green bar, googly eyes.”
Now why does whiting out the number 1 and drawing a heart over it change the probabilities? We can even imagine that each symbol ball has the original number underneath the symbol but it’s only revealed when the symbol is scraped off. Do the lottery gods somehow know that 6 symbols used to make a sequence and thus discriminate against it?
It’s easy to visualize if you look at it drawbydraw:

On the first draw, each number has a 1 in 48 chance of being drawn. So, 1 is just exactly as likely as any other.

On the second draw, there are 47 balls left, so each one has a 1 in 45 chance. 2 is just as likely as any other–why wouldn’t it be?

Third draw, now only 45 balls left, each with a 1 in 44 chance. 3 has just as much a chance of being drawn as another.
…etc. Ask your GF why she thinks sequences are less likely than any other set of numbers.
In lotteries, each number is drawn independently of the others, Thus, sequence is irrelevant.
Grrrrrr. I messed up the numbers. But you get the idea.
Assuming that by “sequence” you mean numbers in succession, you’re wrong. Whether your sequence is 1 2 3 4 5 6 or 2 9 17 21 34 42, the probability of that specific sequence coming up is identical.
It’s more likely for the winning sequence to be scattered than contiguous, but for a specific scattered sequence, the probability is the same as for a specific contiguous sequence.
Here’s a mental exercise. Instead of numbers on the balls, put pictures of 48 different Popes. Now, what effect does the ordering of the Popes have on the balls chosen? Was that ordering alphabetical, chronological, or ordered by percentage of body fat? Does it matter? No.
But you can take any collection of 6 numbers, and devise a simple set of rules that will convert them to a set of consecutive numbers. If you choose for example 3,7,19,23,31,37 one possible set of rules would be:
subtract 2 from the 1st number
subtract 5 from the 2nd number
subtract 16 from the 3rd number
subtract 19 from the 4th number
subtract 26 from the 5th number
subtract 31 from the 6th number
That neatly maps the numbers you picked onto 1,2,3,4,5,6,7.
Obviously the set of rules, the mapping, is arbitrary here, yet your girlfriend maintains that one sequence 3,7,19,23,31,37, is more likely to win than the other 1,2,3,4,5,6,7.
Ask her what specific property of the arbitrary mapping it is that increases her odds of winning with the “random” sequence over your odds with the “ordered” sequence.
If she could come up with such a property, she could apply it iteratively to a starting sequence of numbers, and maximize her chances of winning. Maybe even win every time!
That would render the very concept of a random lottery absurd.
QED is correct (as is often the case)  there’s no way for the odds of any ball being chosen to be influenced by what has gone before.
There’s an analogy with coin flips. You can get a lot of people to agree that after a sequence of, say, ten heads in a row, tails is more likely on the next flip. But the coin has no memory  no way to know what has happened and thus how it should be expected to behave. So (with a fair coin) the odds stay at 50/50 no matter what the history of previous flips has been.
nTucker: That’s not what I meant.
You’re right, that as Xema pointed out: “there’s no way for the odds of any ball being chosen to be influenced by what has gone before.”
BUT
What I was talking about is that the odds become greater against a particular sequence the more possible numbers (or balls, dice, etc) you toss into the equation.
If you roll 1 6sided die you have a 16 chance of rolling any particular number. But what happens when you toss 2 dice? the chances of you getting a particular sequence of numbers there (say, two 1 ones) is now much less. And so if you keep adding dice the chances of a particular sequence coming up become less and less likely.
Her reasoning is probably along the lines of “when was the last time you saw a list of sequential numbers drawn as the winners?” This logic falls into the classic trap of anecdotal evidence.
Use a deck of cards to simulate the lottery drawing. Do ten drawings. It is highly unlikely you will draw six sequential cards in any of the ten drawings. But it is equally unlikely to draw any predetermined sequence of numbers.
Q.E.D.: Fine point, but I must confess I was disappointed at your typing errors. I usually regard your mathematical posts as gospel.
Opus1: Brilliant post. If she reads your post and still disbelieves, there is no hope for her. Perhaps she’d be willing to play “Monty Hall” with me for money?
My apologies. My post had been originally based on my (mistaken) understanding that there were 46 numbers, not 48. In correcting, I made several errors.
Ask you girlfriend by what magic the lottery numberpicking machine knows the numbers are consecutive. Be as sarcastic as possible when doing this. Sneer, if you can.
(maybe you should just ignore me:))
once your girlfriend has accepted that what you and everyone else here said is true, ask her how likely she thinks you’d hit the jackpot anyway. !@#$% impossible right? the exact same will apply to her number too.