Answer

Solution

1. 1

   Let $X$ be the total value of two chips. The possible outcomes (unordered):

2. 2

   \begin{itemize}[leftmargin=*]

3. 3
   • Both Re.\ 1: $\binom{3}{2}/\binom{5}{2} = 3/10$, $X = 2$.
4. 4
   • One of each: $\binom{3}{1}\binom{2}{1}/\binom{5}{2} = 6/10$, $X = 5$.
5. 5
   • Both Rs.\ 4: $\binom{2}{2}/\binom{5}{2} = 1/10$, $X = 8$.
6. 6

   \end{itemize}

7. 7
   $$E[X] = 2 \cdot \tfrac{3}{10} + 5 \cdot \tfrac{6}{10} + 8 \cdot \tfrac{1}{10} = \tfrac{6 + 30 + 8}{10} = \tfrac{44}{10} = 4.4.$$
8. 8

   Player pays nothing (gain = receipts) so the expected gain is $4.4$, which lies in $(4,5)$.

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