A whole number between 100 and 999 inclusive is chosen at random. Find the probability that it is exactly divisible by 3. If it is exactly divisible by 3, what is the probability that it is exactly divisible by 9?
[Math] probability question involving numbers 100 to 999 inclusive
probability
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Best Answer
Please do not count...
For each $ab$ in $\{0,1,\ldots,9\}\times\{0,1,\ldots,9\}$ there are $9$ numbers in the set which end by $ab$. Among them, exactly $3$ are multiples of $3$ and exactly $1$ is a multiple of $9$. Thus the probability that a number chosen unformly at random in the set $\{100,101,\ldots,999\}$ is some multiple of $3$ is exactly $\frac13$ and the probability that it is some multiple of $9$ is exactly $\frac19$.
This applies to every set $\{9k+i+1,9k+i+2,\ldots,9\ell+i\}$ with $0\leqslant k\lt\ell$ and $i\geqslant0$.