Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are the same?
- A)
- B)
- C)
- D)
- E)
Answer
D
Key insight
Whatever color Alice sends, Bob's bag holds six balls with that color doubled, and the bags match exactly when Bob sends back one of those two.
Solution
Suppose Alice transfers a ball of some color, say green. Alice now has four colors (no green); Bob has six balls, two of which are green.
For the bags to end up identical, Alice must get a green ball back: then both bags again hold one of each color. If Bob sends any other color, Alice ends up with two of that color and no green while Bob has two greens, so the bags differ.
Bob picks uniformly among his six balls, and two of them are green, so the probability is .
Because this reasoning holds for every color Alice might choose, the overall probability is also .
The answer is .
Why this works
Condition on the first random step; by symmetry among the colors, the situation after Alice's move looks the same regardless of her choice, so the answer is the conditional probability of a single scenario. Only colors matter, not which physical ball moves, which is why the count of favorable balls is , not .
The trap
Requiring Bob to return the exact same ball (1/6), forgetting that Bob's own ball of that color works equally well.
Common mistakes
- Requiring Bob to return the exact same ball (1/6), forgetting that Bob's own ball of that color works equally well.
- Using as Bob's total (forgetting the ball Alice added), giving or .
Techniques
Exploit symmetry to reduce work or pair up objects