Math Is Fun Forum

magniack

Guest

Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

I have 5 dice. I am allowed to throw them twice. My goal is to score at
least 3 '1'. When I throw a '1' the first time I am allowed to put the
die on the side.

Example:
First Throw:   1 3 1 4 6
Second Throw:  - 2 - 1 4 --> score: 3 '1'

I simulated the problem with a program and got a probability of
0.170521563 the first time and the second time I got 0.170500784
(simulated it 10^9 times)
But I when I calculated I got always different results. Here are my
different ideas:

My first try was to calculate the probability of a maximum of 2
'1' (complementary event). That got me the formula:

(but this was too low; 1-0.73569023117982524 = 0.26430976882017476)

So I actually calculated the prob. of the 3 '1'; got me the formula:

(This was too low as well and it didn't even add up to 1. But I
couldn't figure out why. even tough I think I didn't subtract the
dice thrown - so I can't get the correct result. But still it should
add up to 1.)

So I added all the probabilities up of each single throw:
(First throw (number of 1's),Second throw (numbers of 1's))

0,3 ->

1,2 ->

2,1 ->

3,0 ->

0,4 ->

1,3 ->

2,2 ->

3,1 ->

4,0 ->

0,5 ->

1,4 ->

2,3 ->

3,2 ->

4,1 ->

5,0 ->

And the Solution was 0.12891671866267 what is still too low.

I would be very thankful for your help. And sorry for my english but
I am not living in the states anymore. Thanks a lot.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Hi;

I simulated the problem with a program and got a probability of
0.170521563 the first time and the second time I got 0.170500784

Are you saying that is 2 separate simulations? So you are estimating the probability as 17%.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

hi Magniack

I've calculated it and agree with all the simulations.

My calculations are in Excel so the screen shot below shows the formulas I used, and then the probabilities.

I've cropped the column headings.  Whoops.  Column E has the first throw results and G the second.

Bob

Last edited by Bob (2011-01-24 01:53:26)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

That is what I am getting too. Nice work Bob!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Thanks bobbym.   

Did you notice I asked David to post a reply?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Yes, I did. I did not get a chance to say hello.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Obviously, my abstract algebra isn't up to it;  but his is;  it's his specialism; so it seemed appropriate.  And I loved that little touch at the end .... just finish this off   .... he does that to me    .... and three days later .... I think I need another hint please.

He once solved one of those string and rod with holes in puzzles that I'd been struggling with by watching me and just doing it straight off once I stopped moving it about and let him have it!

Not that I'm suggesting I'm Sherlock or anything, that would be vain, but he is definitely my Mycroft.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

I see that you are very proud of him.

There is always someone smarter, better, faster... Bet you trained him. Or at least started him off. Case of the tail wagging the dog. I have a brother like that.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

magniack

Guest

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Hi;

I simulated the problem with a program and got a probability of
0.170521563 the first time and the second time I got 0.170500784

Are you saying that is 2 separate simulations? So you are estimating the probability as 17%.

Sorry about that... yes 2 seperate simulations.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Hi magniack;

I believe post #3 , Bob Bundy's solution is what you are after!

Welcome to the forum!

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

magniack

Guest

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Yes I just saw it. Thanks a lot. that helps...

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Probability:5 dice, throw them twice,get more than 3 '1', take out 1's

Hi magniack,

You can use a tree diagram, it'll be easier to visualize such problems.
The answer is 0.17050733

The exact answer is:

where p = 1/6 and q=5/6, the probability of getting 1, and binomial distribution is used.
Reasoning is 0 1's in first throw, then atleast 3 1's in the second, and similarly for 1 1's, 2 1's so on...

Everything agrees with the simulation.

*edit* I thought bob bundy's excel sheet was simulation, and posted the same solution. But I enjoyed solving, thanks.

Last edited by gAr (2011-01-24 06:12:33)

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."