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a busy cat

Given: a box contains $4$ black cubes, $6$ black spheres, $6$ white cubes and $x$ white spheres.

$${\bf P}(\text{black object}) = \frac{10}{16+x}$$

Now I don't know what to do.

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Hello, could you type in the question? This actually enhances the experience here, where we can search for your question with the words. Your question has a few key words eg. experiment, random, independent. Thanks! – bryansis2010 Mar 13 '13 at 7:04
Just apply the definition of independence and solve for $x$. – Macavity Mar 13 '13 at 7:04

Your experiment has

  • $4$ black cubes
  • $6$ black spheres
  • $6$ white cubes
  • $x$ white spheres
  • with a total of $16+x$ objects

$P(A) = P(\text{ cube drawn })\frac{10}{16+x}$ and $P(B) = P(\text{ black drawn })\frac{10}{16+x}$

If $A$ and $B$ are independent, we can apply the formula $P(A) \times P(B) = P(A\cap B)$

Now, realize that $P(A\cap B) = P(\text{ black cube drawn })= \frac{4}{16+x}$.

We can formulate the following equation:

$$ \frac{10}{16+x} \times \frac{10}{16+x} = \frac{4}{16+x} $$

Solving the equation, you should get $x=9$.

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For these events to be independent, the ratio of cubes to spheres has to be the same for white and black objects; equivalently the ratio of white objects to black objects has to be the same for cubes and spheres. This happens when $x=9$.

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