Let P(i) be the probability that you get a vowel for cube i. So,

P(i)=1/3 for i={1,2,4,5,7,8,9,11,13,14,16}

P(i)=1/2 for i={3,10,12,15}

P(i)=2/3 for i={6}

Let K= the product of (1-P(i)) for i from 1 to 16; which equals (2/3)^11*(1/2)^4*(1/3)^1 [this is the probability of getting no vowels]

For each i let Q(i) = P(i)*K/(1-P(i)) [this is the probability that cube i is the only vowel]

Each Q(i) is mutually exclusive from each other so the total probability of the event of getting only one vowel is the sum of Q(i) from i=1 to 16.

This simplifies to:

16*K*[11*(1/2)+4*(1)+2*(1)]=11.5K=23*2^6/3^12

This is closer to 1 out of 361. I verified this result with quick spreadsheet.

Now, looking at your code I see a lowercase y in your vowels and uppercase Y in your die. If it is case sensitive I can see the probability being lower than it should be.

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