Problem 6 · Complete the random order and use symmetry · 4 points
Made by AI and not reviewed by a real person.
A bag holds 5 red and 5 blue marbles. You draw without replacement until all the marbles of one color have been drawn. What is the expected number of marbles left in the bag?
- 3/2
- 5/6
- 1
- 5/3
- 2
Show a hint
- Imagine continuing the draws until the bag is empty. How could the end of that complete sequence describe what was left when you stopped?
- Count reds after the last blue and blues after the last red separately. Think about the six gaps around the five marbles of the opposite color.
Show the answer and solution
Answer: D — 5/3
In a complete draw order, the marbles left when you stop are either the reds after the last blue or the blues after the last red. The five blues create six gaps for the reds. By symmetry, each gap contains an expected 5/6 reds, so an expected 5/6 reds follow the last blue. Likewise, an expected 5/6 blues follow the last red. The expected number left is therefore 5/6 + 5/6 = 5/3. The tempting answer 5/6 counts only the cases where reds are left behind.