Anyone good at probability?

[ringo]

Maths help required.

I was doing the lottery and my lucky dip ticket gave me four numbers in the 40s.

My son (13) said this was less likely to win than a more even spread.

I replied that any combination of six numbers had an equal probability of winning. As each was 1/50 - 1/49 - 1/48 etc. (or however many numbers there are in the draw).

He replied but if you have two from the forties (for example) there are only eight remaining, and so the odds are greater for selecting another from the forties. he also added - if there are 49 balls numbered 1 and one ball numbered 40 surely there is not an equal probability of selecting the ball numbered 40 at random?

My brain blew a fuse at this point.

Can someone explain the real position?
Then I can counter/embrace the ridicule currently aimed at me by my son.

Many thanks in advance -and a happy new year.

 

[Bullet]

Your son is wrong

 

[scaramanga]

Your son is wrong

I like to bathe in custard.

In fact, as people tend not to like clustered numbers you increase your probable return (but not your chance of those numbers getting g drawn) with a cluster of numbers.

 

[Richie]

You're right, your son is wrong. The odds for each ball are independent of previous balls, just like if a fair coin comes up heads 10 times in a row the next flip is still 50% chance heads and 50% chance tails. The odds of any ball coming out is unaffected by the previous ball and does not effect the next ball.

I think I know what he's trying to say and I'll explain it with the coin analogy. He is correct that the odds of a more uniform split overall is more likely than having 4 balls from the 40's, but the odds of any six ball combination is the same as any other.

Consider the odds of flipping a fair coin 10 times. It's obvious that getting 10 heads and zero tails is less likely than getting 5 heads and 5 tails, and this is because there a more ways of getting 5 heads and 5 tails than getting 10 heads. To be exact there is only 1 way of getting 10 heads and 252 ways of getting 5 heads and 5 tails, however the odds of getting 10 heads and the odds of any specific combination of 5 heads and 5 tails is the same. This means getting HHHHHHHHHH, HHHHHTTTTT or HTHTHTHTHT all have have the same probability.

To go back to the lottery, there are more ways to get a uniform split of numbers than have 4 come out in the 40's so the odds are better that a uniform split will happen just like you're more likely to get a 50/50 split when flipping a coin than all heads, but the odds of any specific 6 numbers coming out are the same. Hopefully that made sense.

 

[Bullet]

You're right, your son is wrong. The odds for each ball are independent of previous balls, just like if a fair coin comes up heads 10 times the next flip is still 50% chance heads and 50% chance tails. The odds of any ball coming out is unaffected by the previous ball and does not effect the next ball.

I think I know what he's trying to say and I'll explain it with the coin analogy. He is correct that the odds of a more uniform split overall is more likely than having 4 balls from the 40's, but the odds of any six ball combination is the same as any other.

Consider the odds of flipping a fair coin 10 times. It's obvious that getting 10 heads and zero tails is less likely than getting 5 heads and 5 tails, and this is because there a more ways of getting 5 heads and 5 tails than getting 10 heads. To be exact there is only 1 way of getting 10 heads and 252 ways of getting 5 heads and 5 tails, however the odds of getting 10 heads and the odds of any specific combination of 5 heads and 5 tails is the same. This means getting HHHHHHHHHH, HHHHHTTTTT or HTHTHTHTHT all have have the same probability.

To go back to the lottery, there are more ways to get a uniform split of numbers than have 4 come out in the 40's so the odds are better that a uniform split will happen, but the odds of any specific 6 numbers coming out are the same. Hopefully that made sense.

Very nice, you should be a teacher

 

[ringo]

Thanks very much everyone (especially Richie for taking the time to write such a long and simple explanation).

I must admit I still struggle with this bit -

"...there are more ways to get a uniform split of numbers than have 4 come out in the 40's so the odds are better that a uniform split will happen.."
and
"the odds of any specific 6 numbers coming out are the same"

is that saying some combinations are more/less likely than others but the odds of any six numbers are the same?
Because that makes it sound as if I'd be more likely to match the random lottery machine choosing a uniform split than choosing four from the 40s and two from the thirties, for example.

Anyway, I think I'm a lost cause when it comes to this kind of thinking (but thanks for trying Richie) - probably why I'm a writer and not an actuary or computer programmer.

I've got to move on to another challenge now - all I need is someone to teach me how to beat my son at Call of Duty:Black Ops 2!

 

[ringo]

update: had a eureka moment while eating lunch and now get while all combinations are equally likely (i think)

 

[Robbo]

What, no silly word play??

 

[Richie]

Thanks very much everyone (especially Richie for taking the time to write such a long and simple explanation).

I must admit I still struggle with this bit -

"...there are more ways to get a uniform split of numbers than have 4 come out in the 40's so the odds are better that a uniform split will happen.."
and
"the odds of any specific 6 numbers coming out are the same"

is that saying some combinations are more/less likely than others but the odds of any six numbers are the same?
Because that makes it sound as if I'd be more likely to match the random lottery machine choosing a uniform split than choosing four from the 40s and two from the thirties, for example.

Anyway, I think I'm a lost cause when it comes to this kind of thinking (but thanks for trying Richie) - probably why I'm a writer and not an actuary or computer programmer.

I've got to move on to another challenge now - all I need is someone to teach me how to beat my son at Call of Duty:Black Ops 2!

I'll try and explain again using just the lottery as an example, maybe somebody else is having a similar problem and part of me enjoys explaining this kind of thing.

Imagine a lottery where you are choosing 5 numbers from a possible 50. Now lets consider two tickets, (a) getting the first 5 from the 40's [40, 41, 42, 43, 44] or (b) getting the first 4 from the 40's [40,41,42,43] and one from anywhere not in the 40's. Lets say we get really lucky and the first 4 balls are 40, 41, 42, 43 and we're waiting on the last ball. For option (a) we have to get a specific ball to come out, 44, whereas for option (b) we've got the 4 that we wanted in the 40's and now any ball outside the 40's will get us this result (i.e. not 44,45,46,47,48,or 49). After those first 4 balls, option (a) has a 1/46 probability of winning and option (b) has a 40/46 probability of winning which is obviously more likely.

Of course when you play the lottery you don't bet on ranges of numbers you bet on specific numbers. You aren't able to place the bet (b) because you can't just say 'one from not in the 40's', you have to choose a specific number, so lets look at it again. Keep option (a) the same and choose the last number of option (b) to a specific numbers, so your ticket for (b) becomes [40,41,42,43,1]. I've chosen 1 to be my 'not in the 40's' number), we play again and again the first 4 numbers are 40,41,42 and 43. Now ticket (a) again has 1/46 odds of winning because we need a 44, and option (b) also has 1/46 odds because we need a 1.

Basically you can't bet on what won't come up or ranges of what will come up, you have to bet on a specific number. I have a feeling I might have made it less clear but hopefully it makes sense.

 

[braineclipse]

I like to bathe in custard.

In fact, as people tend not to like clustered numbers you increase your probable return (but not your chance of those numbers getting g drawn) with a cluster of numbers.

Yes. Apart from a couple of clusters including 1,2,3,4,5,6 apparently.

Choosing numbers over 30/31 also gives you an increased probable return as people like picking numbers matching birthdays etc.

I'll try and explain again using just the lottery as an example, maybe somebody else is having a similar problem and part of me enjoys explaining this kind of thing.

Imagine a lottery where you are choosing 5 numbers from a possible 50. Now lets consider two tickets, (a) getting the first 5 from the 40's [40, 41, 42, 43, 44] or (b) getting the first 4 from the 40's [40,41,42,43] and one from anywhere not in the 40's. Lets say we get really lucky and the first 4 balls are 40, 41, 42, 43 and we're waiting on the last ball. For option (a) we have to get a specific ball to come out, 44, whereas for option (b) we've got the 4 that we wanted in the 40's and now any ball outside the 40's will get us this result (i.e. not 44,45,46,47,48,or 49). After those first 4 balls, option (a) has a 1/46 probability of winning and option (b) has a 40/46 probability of winning which is obviously more likely.

Of course when you play the lottery you don't bet on ranges of numbers you bet on specific numbers. You aren't able to place the bet (b) because you can't just say 'one from not in the 40's', you have to choose a specific number, so lets look at it again. Keep option (a) the same and choose the last number of option (b) to a specific numbers, so your ticket for (b) becomes [40,41,42,43,1]. I've chosen 1 to be my 'not in the 40's' number), we play again and again the first 4 numbers are 40,41,42 and 43. Now ticket (a) again has 1/46 odds of winning because we need a 44, and option (b) also has 1/46 odds because we need a 1.

Basically you can't bet on what won't come up or ranges of what will come up, you have to bet on a specific number. I have a feeling I might have made it less clear but hopefully it makes sense.

If 40, 41, 42, 43 and 44 have all come out...

Your kid is right, in his assumption, but the conclusion doesn't follow. A ball from 1-10 is more likely than a ball in the 40s. But any specific ball 45+ is no more likely than a specific ball in the 1-10 range. 45 is no less likely than 7. And since you have to pick specific numbers all picks have the same probability.

 

[Mr_B]

if you're at a roulette table and red comes up 8 times in a row, you'd think, ok time to bet on black. No, its just a random sequence, its still 50-50 to be red or black on the next spin.

 

[ringo]

Thanks everyone for taking the time to explain - I do get it properly now. As does my son (after some heated discussion). If only my maths teacher had been as patient, clear and concise. Once you get into it it's actually quite fun - (not that I'll ever admit to it in public). Although I realise this is the simple stuff - it would be fascinating to appreciate the world through the mind of Nash or Von Neumann for a bit.

 

[milo]

Thanks everyone for taking the time to explain - I do get it properly now. As does my son (after some heated discussion). If only my maths teacher had been as patient, clear and concise. Once you get into it it's actually quite fun - (not that I'll ever admit to it in public). Although I realise this is the simple stuff - it would be fascinating to appreciate the world through the mind of Nash or Von Neumann for a bit.

I've recommended this book on here before but it is a really good if this discussion has pricked your interest

 

[ringo]

I've recommended this book on here before but it is a really good if this discussion has pricked your interest

I'll check it out, thanks. [The link didn't show up in my browser, but I got it from the code] I interview a lot of finance academics, might help me make head or tail of what they say.

 

[AuroRaman]