Two twenty-sided dice showing the same face

Question

You roll two fair 20-sided dice, one red and one blue. Each die shows a number from 1 to 20, all equally likely, and the rolls are independent. What is the probability that both dice show the same number?

Accepted Answer

Look at the red die first; it shows some number, say r. That value is now fixed. The blue die is equally likely to land on any of its 20 faces, and only one of those faces (r) counts as a match. So P(	ext{same face}) = \frac{1}{20} = 0.05, which is 5\%. It does not matter which number the red die showed. The moment one die is settled, the second die matches it with probability 1/20. Takeaway. Any two draws from d equally likely outcomes match with probability 1/d. Here d = 20, so the answer is 1/20 = 5\%.

Two twenty-sided dice showing the same face

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