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Here’s another probability question.

Your mate has a proposition.
You choose a number between 1 and 6.
You then roll three fair dice. If your number comes up he’ll pay you a quid, if not then you pay him a quid.

A fair game, right? You can choose any one of six numbers but with three dice you have a fifty-fifty chance of hitting your number.

Your mate see’s you’re hesitant, so he sweetens the deal. If your number comes up on two of the dice he’ll pay you double (£2).

Not enough? If all three dice show your number he’ll pay you triple (£3).

Your mate is practically a charity!

Should you play his game? If not, why not?
 
btiw2 said:
You’re frighteningly good at this.

It’s difficult for me to choose a favourite mathematician, but John Conway would make my “top ten”.

He didn’t get the next sequence, but once it was explained to him he performed detailed analysis of the sequence. Don’t feel bad if you don’t get it, but don’t google it either.

I’ll present the raw sequence and then add some punctuation to help.

1,11,21,1211,111221,312211,13112221,....

That’s the sequence where the genius mathematician Conway couldn’t determine the next term (intimidated yet?).

No advanced maths is needed. A child could write the next term.

With punctuation and spacing to help see the pattern. Each term on a new line, commas and semicolons added for readability.
1
1,1
2,1
1,2;1,1
1,1;1,2;2,1
3.1;2,2;1,1
1,3;1,1;2,2;2,1

What’s next?
Click to expand...


Between eating dinner late, scratching my head and trying to figure it out, my guess is 11122132 with me losing my way halfway through the latter part of the sequence. I'm perfectly happy to have the errors of my ways explained!
 
Jord86 said:
Between eating dinner late, scratching my head and trying to figure it out, my guess is 11122132 with me losing my way halfway through the latter part of the sequence. I'm perfectly happy to have the errors of my ways explained!
Click to expand...
Here’s the sequence again.

1,11,21,1211,111221,312211,13112221,....

Let’s look at the first term.

What do we have?
1

But the digits repeat in later terms. What do we have in the first term?

One 1.

Does that help?
 
btiw2 said:
Here’s the sequence again.

1,11,21,1211,111221,312211,13112221,....

Let’s look at the first term.

What do we have?
1

But the digits repeat in later terms. What do we have in the first term?

One 1.

Does that help?
Click to expand...


Not much unfortunately, I understood your tiered/pyramid interpretation better, 1,1,2- 1,1,3 etc could see a correlation forming there.
 
btiw2 said:
Here’s another probability question.

Your mate has a proposition.
You choose a number between 1 and 6.
You then roll three fair dice. If your number comes up he’ll pay you a quid, if not then you pay him a quid.

A fair game, right? You can choose any one of six numbers but with three dice you have a fifty-fifty chance of hitting your number.

Your mate see’s you’re hesitant, so he sweetens the deal. If your number comes up on two of the dice he’ll pay you double (£2).

Not enough? If all three dice show your number he’ll pay you triple (£3).

Your mate is practically a charity!

Should you play his game? If not, why not?
Click to expand...


Aaagh I can't explain what I mean adequately...... No because your mate can't lose, and it's like an accumulator (well as I see it), the odds of you hitting your number on the trot are multiplied each time becoming rarer to achieve. Something like that anyways.
 
Jord86 said:
Aaagh I can't explain what I mean adequately...... No because your mate can't lose, and it's like an accumulator (well as I see it), the odds of you hitting your number on the trot are multiplied each time becoming rarer to achieve. Something like that anyways.
Click to expand...
That’s a horrible feeling. You know something but can’t explain it. It’s like when someone asks you to define a word you know.

Maths is like that. When you’re young maths is about finding the answer (x=...).

Later, it becomes “show that...” or “prove that...”. They literally give you the answer, but the job becomes telling the story of why something is true. A much harder proposition.

It’s harder to tell a story why it’s a bad bet (and you’re right, it is).

I’ll wait and see if DA offers an explanation before I offer my proof.
 
dwlondon said:
The 'mate' isn't choosing his number either and has 5 in 6 chance of winning.
Click to expand...
But you roll three dice.

Say you bet on 6. If the dice came up 123 then you pay him a quid.
If it was 2,3,6 he pays you a quid.
If it was 2,6,6 he pays you two quid.

If the dice are all different then half the numbers pay out - a fifty/fifty bet. Half the time you win, half you lose. Like a coin toss.

If you get two or three matches then you make even more. So why’s it a bad bet?
 
btiw2 said:
So it’s
1
We have one 1.
Let’s write that down. We have 11
...
Click to expand...


Right I'm being perfectly truthful I had to look up John Conway to gain an understanding of the sequence, otherwise I'd have chewed through my arms by now, but I honestly didn't check the answer, just the method, so I still may be wrong: 1113213211.

Infuriatingly simple idea, yet confusing and brilliant at the same time, very clever mixing the word of the number with the actual numbers together, fools the brain into thinking pure number logical sequence, rather than counting whilst you're going. If that makes any sense.
 
The odds are against the gambler as always.He needs to be convinced he might just win.

I think the probability is 1/6^n n being number of dice.

But await mathematical proof.
 
Jord86 said:
Right I'm being perfectly truthful I had to look up John Conway to gain an understanding of the sequence, otherwise I'd have chewed through my arms by now, but I honestly didn't check the answer, just the method, so I still may be wrong: 1113213211.

Infuriatingly simple idea, yet confusing and brilliant at the same time, very clever mixing the word of the number with the actual numbers together, fools the brain into thinking pure number logical sequence, rather than counting whilst you're going. If that makes any sense.
Click to expand...
Maddening, isn’t it?

Can you imagine how hard Conway found it? I’m sure he considered all sorts of abstruse mathematics- yet a four-year old could have told him the answer.

FWIW I don’t think I worked it out when I first saw it.
 
dwlondon said:
The odds are against the gambler as always.He needs to be convinced he might just win.

I think the probability is 1/6^n n being number of dice.

But await mathematical proof.
Click to expand...
Okay. Here’s what is called a “moral proof”. It explains why it’s true but isn’t algebraically rigorous.

Imagine I bet on every number.

I roll the dice and get three different numbers. Say 4,5,6
I lose a quid on 1,2 and 3
But I win a quid on 4,5 and 6
No effect. I lose three quid and win three quid. Evens out.

But say I’d rolled 5,6,6
I’d lose on the 1,2,3 and 4
I’d win on the 5 and win double on the 6
I’d lose £4 and win £3. Overall loss of £1.

Say I rolled 6,6,6
I’d lose on the 1,2,3,4,5
Win treble on the 6
I’d lose £5 and win £3. Overall loss £2

If you go through the algebra, the expected payout is about 94-95% (from memory). Better than a fruit machine or a bookie, but not a good bet.
 
btiw2 said:
Okay. Here’s what is called a “moral proof”. It explains why it’s true but isn’t algebraically rigorous.

Imagine I bet on every number.

I roll the dice and get three different numbers. Say 4,5,6
I lose a quid on 1,2 and 3
But I win a quid on 4,5 and 6
No effect. I lose three quid and win three quid. Evens out.

But say I’d rolled 5,6,6
I’d lose on the 1,2,3 and 4
I’d win on the 5 and win double on the 6
I’d lose £4 and win £3. Overall loss of £1.

Say I rolled 6,6,6
I’d lose on the 1,2,3,4,5
Win treble on the 6
I’d lose £5 and win £3. Overall loss £2

If you go through the algebra, the expected payout is about 94-95% (from memory). Better than a fruit machine or a bookie, but not a good bet.
Click to expand...


That explanation has just lifted a small part of the brain freeze I've had since the first problem you posted earlier tonight :D it's the most challenging Saturday evening I've had in quite a while!
 
Jord86 said:
That explanation has just lifted a small part of the brain freeze I've had since the first problem you posted earlier tonight :D it's the most challenging Saturday evening I've had in quite a while!
Click to expand...
Thank you, but DA started this trend to offer puzzles.

As a point of trivia. At school I played all sorts of card and dice game. My trousers were down my thighs with the change in my pockets.

Timid students hesitated to play poker with me, but dice? Where they can pick the number they’re betting on? They played that.

The game outlined here was a good share of my pocket money.

It’s a deceptive game. 5% edge doesn’t seem much. But over the long term (and dice is a fast game - lots of rolls per minute) it adds up. I just had to make sure I had enough money in my pockets to be the bank until their luck changed.

I don’t think it’s an exaggeration to say that the majority of my maths education stemmed from a desire to fleece my classmates.

Try it. Give your missus/kids a load of pennies and play the bank. The bank doesn’t always win. It’s not a con. It feels fair (ha!). It’s more similar to the percentage of a roulette wheel. And maybe your missus or children will feel like learning about probability theory afterwards.
 
btiw2 said:
Thank you, but DA started this trend to offer puzzles.

As a point of trivia. At school I played all sorts of card and dice game. My trousers were down my thighs with the change in my pockets.

Timid students hesitated to play poker with me, but dice? Where they can pick the number they’re betting on? They played that.

The game outlined here was a good share of my pocket money.

It’s a deceptive game. 5% edge doesn’t seem much. But over the long term (and dice is a fast game - lots of rolls per minute) it adds up. I just had to make sure I had enough money in my pockets to be the bank until their luck changed.

I don’t think it’s an exaggeration to say that the majority of my maths education stemmed from a desire to fleece my classmates.

Try it. Give your missus/kids a load of pennies and play the bank. The bank doesn’t always win. It’s not a con. It feels fair (ha!). It’s more similar to the percentage of a roulette wheel. And maybe your missus or children will feel like learning about probability theory afterwards.
Click to expand...


Years ago a friend of mine studying maths at Bath University managed to subsidise the majority of his first years costs by participating in online casinos in Blackjack. More accurately, he would sign up, be given £50-£100 of free bets on average, then on the basis of probability would play X amount of hands on the lowest blind amount until the stipulation of certain amount of games played ran out, then would transfer his winnings/credit out of his account and close it down, to move onto the next online casino. He was regimental in sticking to a routine to twist on 16 and under or stick on 17 or over (I think). Either ways, he'd play in some cases a couple thousand games per casino with no thought or emotion, just sticking to his cut off points on probability, then cutting and running. After about 4 months he'd amassed around £3000, just about enough to have covered his bar bill at the Students Union!

Ironically he ended up getting a Third, so, worse than if he hadn't bothered going to Uni, retrained and took dozens of exams and is now plying his trade as an Actuary.......
 
btiw2 said:
Here’s the sequence again.

1,11,21,1211,111221,312211,13112221,....

Let’s look at the first term.

What do we have?
1

But the digits repeat in later terms. What do we have in the first term?

One 1.

Does that help?
Click to expand...

Yes, that helps.

I got it :p

Honestly.
 
btiw2 said:
Thank you, but DA started this trend to offer puzzles.

As a point of trivia. At school I played all sorts of card and dice game. My trousers were down my thighs with the change in my pockets.

Timid students hesitated to play poker with me, but dice? Where they can pick the number they’re betting on? They played that.

The game outlined here was a good share of my pocket money.

It’s a deceptive game. 5% edge doesn’t seem much. But over the long term (and dice is a fast game - lots of rolls per minute) it adds up. I just had to make sure I had enough money in my pockets to be the bank until their luck changed.

I don’t think it’s an exaggeration to say that the majority of my maths education stemmed from a desire to fleece my classmates.

Try it. Give your missus/kids a load of pennies and play the bank. The bank doesn’t always win. It’s not a con. It feels fair (ha!). It’s more similar to the percentage of a roulette wheel. And maybe your missus or children will feel like learning about probability theory afterwards.
Click to expand...

Is there only a 5% edge using that 3-dice trick?
 
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