Problem 29 · 2009 Math Kangaroo
Stretch
Counting & Probability
careful-countingcomplementary-counting
Samantha and her three sisters go to the theatre. They have reserved a loge with four seats. Samantha and two of her sisters arrive early and sit down without paying attention to their seat numbers. Marie arrives later and insists on sitting in the seat indicated on her ticket. What is the probability that Samantha has to change her seat, if every sister who has to swap seats then insists on sitting in the seat indicated on her ticket?
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Answer: B — 12
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Hint 1 of 2
The first three sit at random; then Marie’s arrival may start a chain of forced swaps.
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Hint 2 of 2
Work out, over the random seatings, how often that chain ends up moving Samantha.
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Approach: trace the swap chain over all random seatings
- The three early sisters occupy three of the four seats at random; Marie then claims her own seat, displacing whoever is there into a chain.
- Checking every equally-likely seating, Samantha ends up having to move in exactly half of them.
- So the probability is 1/2.
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