Please verify my solution to this problem.
Each of m urns contains three red chips and four white chips. A total of r samples with replacement are taken from each urn. What is the probability that at least one red chip is drawn from at least one urn?
Here is my solution:
$P$(at least one red chip is drawn from at least one urn)$=$
$1-P$(no red chip is drawn from any urn)$=$
$1-P$(only white chips are drawn from the urns)$=$
There is a $\tfrac47$ chance of drawing a white chip out of any one of the m urns.
Since it is replaced, there is an $(\frac47)^{r}$ chance of drawing a white chip for each of the r draws from any urn.
Therefore, $P$(draw at least one red chip from at least one urn) $=$ $1-(\frac47)^{m*r}$
Your working is fine.
The complement event is we do not see any red chips, which means that we only get white chips in all of the $m\cdot r$ trials.
Hence, the answer is $$1-\left( \frac47\right)^{m\cdot r}$$