A good scientific ___ can be repeated by someone else and the same results will be found.

At f = 100 MHz, show that silver $\left(\sigma=6.1 \times 10^{7} \mathrm{S} / \mathrm{m}, \mu_{r}=1, \varepsilon_{r}=1\right)$ is a good conductor, while rubber $\left(\sigma=10^{-15} \mathrm{S} / \mathrm{m}, \mu_{r}=1, \varepsilon_{r}=3.1\right)$ is a good insulator.

In a box, Lynn has b batteries of which d are dead. She tests them randomly and one by one. Every time that a good battery is drawn, she will retum it to the box; every time that a dead battery is drawn, she will replace it by a good one.

(a) Determine the expected value of the number of good batteries in the box after n of them are checked.

(b) Determine the probability that on the nth draw Lynn draws a good battery. Hint: Let $X_n$ be the number of good batteries in the box after n of them are checked. Show that

$$E\left(X_n \mid X_{n-1}\right)=1+\left(1-\frac{1}{b}\right) X_{n-1} \text {. }$$

Then, by computing the expected value of this random variable, find a recursive relation between $E\left(X_n\right)$ and $E\left(X_{n-1}\right)$. Use this relation and induction to prove that

$$E\left(X_n\right)=b-d\left(1-\frac{1}{b}\right)^n .$$

Note that $n$ should approach $\infty$ to get $E\left(X_n\right)=b$. For part (b), let $E_n$ be the event that on the nth draw she gets a good battery. By conditioning on $X_{n-1}$ prove that $P\left(E_n\right)=E\left(X_{n-1}\right) / b$.