Math Is Fun Forum

T1kso

Member

Registered: 2021-03-15

Posts: 1

Help with a dice problem!

Hi there,

Does anyone know how to work out if results of a dice throw experiment confirm a dice is bias?

It’s rolled 60 times.

Here are the results -

1/5
2/9
3/7
4/8
5/14
6/17

Is the dice bias?

My sons teacher says it is my son is adamant it isn’t or at least can’t be proven! He’s autistic so it’s become a big deal...need help!

Thanks

Btw he’s 12

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: Help with a dice problem!

hi T1kso

Welcome to the forum.

Here's how a mathematician would attempt to answer this question.

(1) Build a model that 'describes' the situation.

(2) Use the model to calculate the probability of these results.

(3) Make a decision about whether that probability suggests the dice is biased.

There are several models; I decided to shrink the table to just two possibilities; throw a 5 or 6 and throw something else.

That gives number of 5,6 results = 31 and something else results 29.

This allows me to construct a binomial model for the dice, ie. there are just two types of result.

https://www.mathsisfun.com/data/binomia … ution.html

I need to know the expected result which is n times p where n is the number of trials and p is the probability of getting a 5 or 6 with an unbiased dice.  n = 60, p = 1/3 so np = 20.  This is also called the mean result. This is how many you'd expect to get.  This is sometimes referred to as 'the law of averages'.  Of course, in the real world the dice doesn't always do what is expected so what we want to find out is how likely is it to get a result of 31.

There is a formula for working this out but it involves a lot of calculations, so, in practice, mathematicians use an approximation.  It can be shown that the binomial probabilities approximately follow a normal distribution curve with the same mean and probabilities for the normal distribution have been tabulated, making it easy to look up an answer.

https://www.mathsisfun.com/data/standar … table.html

You also need to know the standard deviation of the binomial model. This is a measure of how much variation there might be if you kept repeating the experiment over and over.  The probability of not getting 5 or 6 is q = 2/3 , and the sd. formula is √ (npq) so the sd. is square root of (60 times 1/3 x 2/3) = √ (40/3)

When using the normal distribution you have to change the expected value and sd. to 'standard' ones so that the look up works.  Otherwise you'd have to have a new table for every mean and sd.  The table on the above page is set up for a mean of 1, and a sd. of 1.

The conversion consists of subtracting the mean from the experimental result and then dividing by the sd.  This is called the Z value.

On the normal curve page there is a graph showing the standard normal curve with mean = 1 and sd. = 1.  It can be used to get the probability of from 0 to Z  or the probability up to Z or (the one we want) the probability of over Z.  So choose the third option.  Then move the mouse to change the Z value.  The bit of the graph above the Z value tells you how likely this result is.  If you slide across until Z = 3.01 you'll see the probability comes out as 0.13%.  That means that the chance of getting that result from an unbiased dice is very very small.  It could happen but it's very unlikely. So I'd say the dice probably is biased.  Sorry for the bad news.

That's a lot to take in so your son might like to explore the binomial and normal distributions a bit to get the hang of it.  He is very welcome to post back here if he has more questions.

Bob

ps.  I've just had a play with the Quincunx simulation here:

https://www.mathsisfun.com/data/quincunx.html

The simulation doesn't allow n to be as high as 60.  15 is the max.  But your son could scale the data down by saying n = 15 (=60/4) and the number obtained by experiment is 8 (roughly 31/4)

Set the left/right to 67%/33% and let the balls drop. Each ball has a 1/3 chance of going right so after 15 'bounces' we have a tally of where the ball ends up each time as if we'd done the dice throwing 15 times and counted the 5-6 results..  A bar graph builds up showing how likely each result is.  He'll see that 5 (the expected result) comes up most often and 8 or above is fairly rare.  That's what suggests the dice is biased.  While I've been typing this I let the quincunx roll.  I've just stopped it at 1001 (I intended 1000 but I wasn't quick enough ).  The total for 8 or above was 81 for me.  81 out of 1000 isn't very high.

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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