$10\%$ of new businesses fail within the first year. The records of new businesses are examined until three businesses that failed within the first year are found. Let $X$ be the total number of businesses examined BEFORE finding three businesses that failed within the first year. What is $P(X \notin [6, 10])$?
I've been staring at this problem trying to figure out if I am doing this right or not. I have:
$X~NB(3,0.1)$
$P(X < 6 \lor 10 < X)= 1-[P(X\le10)-P(X\le5)]$
Would this be correct?
Yes.
$\mathsf P(\{6\leq X\leq 10\}^\complement)~{=1-\mathsf P(6\leq X\leq 10)\\ = 1-\mathsf P(X\leq 10)+\mathsf P(X\leq 5)}$
PS: Though the complement of "$\{6\leq X\leq 10\}$" is "$\{6\gt X\}{~\text{or}~}\{X>10\}$" rather than "and".