When coin $A$ is tossed it comes up heads with probability $\frac{1}{4}$, whereas coin $B$ comes up heads with probability $\frac{3}{4}$. Suppose one of these coins is randomly chosen and is tossed twice. If both tosses come up heads, what is the probability that coin $B$ was tossed? Justify your answer.

In each draw of a lottery, an integer is picked independently at random from the first $n$ integers $1,2, \ldots, n$, with replacement. What is the probability that in a sample of $r$ successive draws the numbers are drawn in a non-decreasing sequence? Justify your answer.