Red-Red Blocks

Probability Level 5

A toy factory creates cube blocks in which each side is randomly colored red or blue. You have all 10 distinct (up to rotation) cubes placed in a bag.

You reach into the bag and pick up one of these blocks, noticing that the one (and only one) side that you see is red. What is the probability that the opposite side is also red?

Strictly between

0

and

\dfrac{1}{ 2}

Exactly

\dfrac{1}{2}

Strictly between

\dfrac{1}{2}

and

\dfrac{2}{ 3}

Exactly

\dfrac{2}{3}

Strictly greater than

\dfrac{2}{ 3}

1 solution

Calvin Lin Staff
Sep 30, 2016

We list out the 10 cubes, and count the different types of face-pairs that there are:

Face types	0 red	1 red	2 red	2 red	3 red	3 red	4 red	4 red	5 red	6 red	Total
			adjacent	opposite	vertex	line	adjacent	opposite
R-R	0	0	0	2	0	2	2	4	4	6	20
R-B	0	1	2	0	3	1	2	0	1	0	10
B-R	0	1	2	0	3	1	2	0	1	0	10
B-B	6	4	2	4	0	2	0	2	0	0	20

Since there are 20 (R-R) pairs and 10 (R-B) pairs, the likelihood that given the first face is red, that the opposite face is also red, is $\frac{20}{20+10} = \frac{2}{3}$ .

Note: You can see how important the prior distribution is. If we allowed for all $2^6 = 64$ possible cubes, then the answer would be $\frac{1}{2}$ .

Yes, the wording of the problem is really critical. Thumbs up now.

Michael Mendrin - 4 years, 8 months ago

Red-Red Blocks

1 solution

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