Question

A bag consists of $10$ balls each marked with one of the digits $0$ to $9$. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit $0$?

Answer 1

Let $X$ denote the number of balls marked with the digit 0 among the 4 balls drawn. Since the balls are drawn with replacement, the trials are Bernoulli trials.
$X$ has a binomial distribution with $n=4$ and $p=\frac{1}{10}$
$\therefore q=1-p=1-\frac{1}{10}=\frac{9}{10}$
$\therefore P(X=x){=}^n{C}_x{q}^{n-x}\cdot p^x$, $x=1,2,...n$
${=}^4{C}_x{\left(\frac{9}{10}\right)}^{4-x}\cdot {\left(\frac{1}{10}\right)}^x$
$P\left(nonemarkedwith0\right)=P(X=0)$
${=}^4{C}_0{\left(\frac{9}{10}\right)}^4\cdot {\left(\frac{1}{10}\right)}^0$
$=1\cdot {\left(\frac{9}{10}\right)}^4$
$={\left(\frac{9}{10}\right)}^4$