Statistics question:

[kj_steroids]

Say you have a container with 10,000 marbles - 9,999 of them being blue, 1 being orange.

Assuming complete randomness: You take a marble, make note of its color, and then put it back into the container. And you do this 10,000 times. What is the percent chance that, after you make those 10,000 picks, you've picked the orange marble at least one

Comments

[the_darai]

hmm, percentage eh? 1/10,000 which would translate to 0.01% being orange. i think i'm right. o.o

[hope_n_forever]

Um...
Let's assume the regular event of the probability of getting the orange marble once is p(A).
Just like the_darai mentioned, p(A) is 1/10,000 or 0.0001 (which is actually .000001%).
However, that's not the final answer!

You had conditioned that you have to draw the marbles 10,000 times with replacement! So, what is the probability of getting the orange marble at least once in 10,000 draws?

This scenario is assumed to be an "AND" one, since we are after the orange marble several times.
So, the equation should be: p(A) * p(A) * p(A) * p(A)... (another 9996 p(A)s)
Let's make this easier by saying the equation is: p(A)^10,000 (p(A) to the power of 10,000)
So, since p(A) = 0.0001, then the equation is: 0.0001^10,000
This equals to 1.E-40,000 , which is 0.0000...(another 9995 zeros) 1

Ergo, the probability of getting an orange marble after 10,000 draws, with replacement, is 1.E-40,000 or 1.E-40,002 %.

Wooh! With 9,999 blue marbles, it’s quite not surprising that the chance of getting the one orange one is sooooooooooo very low!

[kj_steroids]

Wow, good work-up!

That is however the probability of picking the orange marble all 10,000 times. =p

[hope_n_forever]

Oh fudge! I used to be very good at this!
Let me try again...

I think I have to use the 'One Formula', where the probability of ALL events occuring is always equal to one.
Therefore, if we ought to find the probability of getting the orange marble at least once, then we first have to find the probability of NOT getting any orange marbles!

So, the probability of NO orange marbles is p(B), which is 9,999/10,000 or .9999!
Then, let's use the same formula of 'AND', where the equation is p(B) ^ 10,000 = .9999 ^ 10,000 = 0.36786104643292992411133567221851...

So, the REAL final answer is: 1 - (0.9999 ^ 10,0000) = 0.63213895356707007588866432778149...
Ergo, 0.0063213895356707007588866432778149...% ~ 0.0063%

So, I guess I got it right now?

[kj_steroids]

Your math is spot on! However, in determining percent, you'd carry the decimal two spots to the right, giving you a 63.2%. That is the chance to find the orange marble.

I was surprised at first finding out it was roughly a 63% chance to find the orange marble. I had personally assumed the chances would be be much, much higher. Like at least 90%, haha.