This is from Markowitz's Risk-Return Analysis: The Theory and Practice of Rational Investing (Volume One) Chapter 1.
There are 3 lottery tickets one can choose from:
\begin{array} {|r|r|}\hline Alternatives & Chances & Outcome \\ \hline A & 1/1000 & $1000 \\ \hline & 999/1000 & $0 \\ \hline B & 1/100 & $100 \\ \hline & 99/100 & $0 \\ \hline C & 1/2000 & $1000 \\ \hline & 1/200 & $100 \\ \hline & 1989/2000 & $0 \\ \hline \end{array}
And the formula for computing the probabilities of final outcomes when a coin is flipped between lotteries A & B yields
$$P_{$1000} = \frac{1}{2}\frac{2}{2000} = \frac{1}{2000} $$ $$P_{$100} = \frac{1}{2}\frac{20}{1000} = \frac{10}{1000} $$ $$P_{00} = 1 - P_{$1000} - P_{$100} $$ $$ = \frac{1989}{2000} $$
I'm having trouble understanding this. So there are 3 different probabilities here. The first one, $P_{$1000}$ is the probability of the outcome being $1000$ dollars, which is (the probability of getting lottery A from the coin flip) x (the probability of the outcome being $1000$ dollars). That makes sense from the table above. But the second one, $P_{$100}$, the probability of the outcome being $100$ dollars should be $\frac{20}{2000}$ or $\frac{1}{100}$. Why then is it $\frac{20}{1000}$???
You have some typos, and this might be clearer
$$P_{\$1000} = \frac{1}{2}\frac{1}{1000} = \frac{1}{2000} $$ $$P_{\$100} = \frac{1}{2}\frac{1}{100} = \frac{1}{200}= \frac{10}{2000} $$ $$P_{00} = 1 - P_{\$1000} - P_{\$100} \\=\frac{2000}{2000}-\frac{1}{2000}-\frac{10}{2000} \\= \frac{1989}{2000} $$