Units digit of a random product

EasyProbability Combinatorics~4m

Four numbers are drawn independently and uniformly (with repetition) from $\{1, 2, \dots, 9\}$ . What is the probability that the units digit of their product is $1$ , $3$ , $7$ , or $9$ ?

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Units digit in $\{1,3,7,9\}$ means the product is coprime to $10$ (odd and not a multiple of $5$ ).
That requires every factor coprime to $10$ : which of $\{1,\dots,9\}$ qualify? Then it's $(4/9)^4$ .

Answer

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0.039 (± 0.001)

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Units digit of a random product

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