Probability of being selected twice in a week given a set of n people?

Question

Let's say a child is selected out of a group of 10 students each day to stay after school and help clean the classroom. What is the probability that a particular child is selected exactly twice during the same 5-day week? We assume that the students are selected uniformly at random, with replacement.

Answer 1

You can use the Probability mass function, that is the probability of $k$ successes in $n$ trials with $p$ likelihood of occurring is equal to: $$\binom{n}{k}p^k(1-p)^{n-k}$$ We know that $p= \frac{1}{10}, \space n=5$ and $k=2$, so $$\binom{5}{2}(.1)^2(1-.1)^{3} \\ = \frac{5!}{2!(5-2)!}\left(\frac{1}{10}\right)^2\left(\frac{9}{10}\right)^3 \\ = 10 \cdot \frac{9^3}{10^5} \\ =\frac{9^3}{10^4} \\ =7.29\%$$ assuming I crunched the numbers right. At the very least, the probability mass function should be the thing you need to calculate the exact value. If you have a TI-83 or higher, you can use the "Binomprob" function too.