probability (with replacement) (1 Viewer)

[100percent]

There are 5 different coloured marbles in a bag.

i) what is the probability that with 5 random draws (with replacement), you draw all the different coloured marbles?

ii)what is the probability that with 6 random draws (with replacement), you draw every coloured marble atleast once?

iii)what is the probability that with N random draws (with replacement), you draw every coloured marble atleast once?

 

[webby234]

i) 1/5*1/4*1/3*1/2=1/120
ii) (1/120)*1/5=1/600
iii) (1/120)*(1/5)ⁿ

Are you sure? Your answers don't seem to make sense - why would there be less chance of every marble being drawn at least once when you draw more marbles?

I would say the answers are
i) 1(because you can choose any marble first) * 4/5 (because you can then choose 4 of the remaining 5) * 3/5 * 2/5 * 1/5
= 12/625

The other two are a bit trickier - I'll come back to them later.

 

[pLuvia]

Are you sure? Your answers don't seem to make sense - why would there be less chance of every marble being drawn at least once when you draw more marbles?

I would say the answers are
i) 1(because you can choose any marble first) * 4/5 (because you can then choose 4 of the remaining 5) * 3/5 * 2/5 * 1/5
= 12/625

The other two are a bit trickier - I'll come back to them later.

Opps, I must of did non-replacement there or something

 

[webby234]

Still don't think your answers make sense - why would the second one have a smaller chance than the first? I think the answer to ii) is 12/625 * 6 but I'm not quite sure - need to get my reasoning right. Not sure about the third one yet - needs a bit more of a think.

 

[webby234]

Oops - first one should be 24/625 - i can't seem to multiply .

OK - my reasoning behind ii)

There are a total of 5⁶ different possibilities with 5 colours and 6 draws.
Of these, allocating each colour a number, you can have
112345
122345
123345
123445
123455
in any order.

There are 6!/2 ways of arranging each possibility, and five different ways, so you get 5 x 6!/2

So divide by 5⁶ and you get 72/625.

Probably an easy way to do it that will make the general one easier - I'll come back to that later.

 

[100percent]

how did you get 6!/2?

 

[pLuvia]

Have you done perms and combs yet? If so there is a rule that if there is a repeating object then you have to divide the total number of arrangements (6!) by the number of repeating objects (2!)

And since you have a chance of drawing out two of the same colours (this is for (ii)) then you have to divide it by 2! which equals to 2

Dammit, forgot all my perms and combs

 

[100percent]

ah ok, makes sense now. thanks