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A cube is painted on all its faces. It is then cut into 64 smaller identical cubes, which are then thoroughly. What is the probability that 2 randomly chosen smaller cubes have exactly 2 coloured faces each?

I have a doubt , there will be 24 identical cubes whose 2 faces will be painted, 8 identical cubes with 3 faces painted, 24 identical cubes with one face painted and 8 identical cubes with no face painted. Now total cases to pick 2 smaller cubes should be 4C2 not 64C2 . correct me if I am wrong

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If your denominator is $4\ choose 2$, what is your numerator? I agree with André Nicolas, and his denominator isn't a factor of $6$. When you say identical, you really mean indistinguishable. We are both considering all the cubes to be different, then counting how many fit the class of two sides painted. It would be like flipping a coin twice and saying there are three different results because HT is the same as TH. There are three results, but they are not all the same probability. – Ross Millikan May 5 '12 at 5:14
I will choose numerator as one and I have chosen denominator 4C2 because there are 4 types of cubes in the box. – Arpit Bajpai May 5 '12 at 6:01
Would that lead you to claim that the chance of picking any given color is $\frac 14$? This is not correct. Put 10 blue marbles and one red marble in a hat, draw 20 times with replacement, and see if you get around 10 reds. – Ross Millikan May 5 '12 at 14:33
up vote 1 down vote accepted

We can also work directly with the probabilities. Pick the cubes a little cube at random, then pick another. The probability that the first cube picked has exactly $2$ red faces is $\frac{24}{64}$. Given that the first cube picked had exactly $2$ red faces, the probability that the second cube picked has exactly $2$ red faces is $\frac{23}{63}$. Thus the probability we get two cubes with exactly $2$ red faces is $\frac{24}{64}\cdot\frac{23}{63}$.

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When there are 24 identical cubes how there are 24 ways to pick it. IT should be one – Arpit Bajpai May 5 '12 at 5:07
Imagine that we have a total of $64$ cubes, with $24$ identical yellow ones, and the rest identical and blue. Put the $64$ cubes in a bag, shake well, and blindfolded pick a cube at random. Is the probability of picking a yellow equal to $1$? – André Nicolas May 5 '12 at 5:11
In the previous comment, shouldn't you ask if the probability to pick a yellow one is $\frac 12$? – Ross Millikan May 5 '12 at 5:16
In his comment, the OP suggested that it "should be one." – André Nicolas May 5 '12 at 5:19

The problem is that picking the different types of cube has different probabilities. It is easier to consider the 64 cubes distinguishable (number them) and then figure out how many of the choices meet your requirement. In this case, it also deals with the fact that picking a 2 face cube the first time changes the probability that the second pick will be a 2 face cube. In fact, my space would be $64 \cdot 63$, not $64 \choose 2$. How many of those are both 2 face cubes?

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How you are making them distinguishable when I said Identical – Arpit Bajpai May 5 '12 at 5:06

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