# Let us have 10 balls in a urn, probability that all the balls in the urn are white if…

Let us have a urn with 10 balls, the balls are either white or black ( we don't know in which proportions ).We extract 4 balls without reintroducing them back and after each extraction the ball is white, meaning that the 4 balls extracted were all white.I need to find the probability that the urn contains only white balls.

My solution ( but I'm not so sure it's correct ):

we have a white balls and 10-a black balls

I started off by naming the events:

B = " urn contains only white balls "

A(i) = " white ball appears i times in n extractions" for our case n=4 and i=4

and I was thought I should use bayes to find my probability:

$$P(B/A_4)=\frac{P(B)*P(A_4/B)}{P(A_4)}$$

$$P(B/A_4)$$- the probability of all the balls in the urn being white when we know that in 4 extractions all the balls were white

$$P(A_4/B)$$- probability that in 4 extractions the balls will be white knowing that the urn contains only white balls ( which is equal to 1)

$$P(A_4)$$-the probability that we will have 4 white balls in 4 extractions

$$P(B)$$-the probability of all the balls being white

And then started my calculus (at least tried):

For $$P(B)$$ I thought that if we extracted 1 ball from the urn the probability of it being white would be $$\frac{a}{10}$$ so that for 10 balls it will be $$\frac{a}{10}^{10}$$ (and here is my question should I have used the hypergeometric distribution here? I mean to calculate the probability that in 10 extractions (we don't introduce them back) all the balls will be white:$$\frac{{{a}\choose{10}}*{{10-a\choose{0}}}}{10\choose{10}}$$ I not sure what should I do here and I would appreciate some help.

For$$P(A_4)$$ I used the hypergeometric distribution $$\frac{{{a}\choose{4}}*{{10-a\choose{0}}}}{10\choose{4}}$$

I would really appreciate some help with this problem and don't hold back on criticism if you see I did some mistakes (I'm new to probabilities) and thank you in advance.

• Fate filled the urn with a random mechanism creating probabilites $p_k$ $\,(0\leq k\leq 10)$ of obtaining exactly $k$ white balls in the urn. Since we don't know anything about the $p_k$ other than $p_k=0$ $\,(0\leq k\leq 3)$ your question cannot be answered. – Christian Blatter Nov 13 '18 at 19:11
• Tell that to my teacher... – The Virtuoso Nov 13 '18 at 19:50