Alan

The probability is (1/3)*(4/12)+(1/3)*(4/6)+(1/3)*(4/6) = 5/9 

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Simply divide by 1,000,000

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Vectors are either skew (in 3-d) or parallel or they intersect.  If they are parallel they are at an angle of zero relative to each other.  If they intersect you can find the angle using the dot product.

I don't know what it means to find the angle between skew vectors.  In general you would need to project them onto the same plane to find an angle, and this could be different depending on which plane you project onto. However, my knowledge of skew vectors is limited so perhaps I'm wrong.

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Here's the exact value:

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See Bertie's answer below.

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Melody, try it with boxes of size 6, 9 and 20.

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13 diamonds, so probability of 1st card being a diamond is 13/52 = 1/4

Given this, there are 51 cards left, 13 of which are spades, so probability that second card is now a spade = 13/51

Combined probability = (1/4)*(13/51) = 13/204 ≈ 0.064

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It must stop somewhere (i.e. total stopping probability = 1) so, 

p_A + p_B + p_C + p_D = 1

3/8  +  1/4  + 2p_C = 1   (because p_D = p_C)

5/8 + 2p_C = 1

2p_C = 3/8

p_C = 3/16

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prob 1st die odd and 2nd die even = (2/3)*(2/3) = 4/9

prob 1st die even and 2nd die odd = (1/3)*(1/3) = 1/9

So probability that sum is odd = 4/9 + 1/9 = 5/9

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