Problem 18 · 2021 Math Kangaroo
Hard
Counting & Probability
careful-counting
A piece of string is lying on the table. It is partially covered by three coins as seen in the figure. Under each coin the string is equally likely to pass over itself one way or the other (i.e. at each crossing either strand is equally likely to be on top). What is the probability that the string is knotted after its ends are pulled?
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Answer: B — \(\tfrac{1}{4}\)
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Hint 1 of 2
Each hidden crossing is independently 'over' or 'under', so list how many equally likely cases there are.
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Hint 2 of 2
Decide which of those cases actually produce a knot when the ends are pulled.
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Approach: count favourable crossing patterns out of all equally likely ones
- Three crossings, each equally likely two ways, give 2³ = 8 equally likely outcomes.
- Only the patterns that interlock the strand into a true knot count; exactly 2 of the 8 do.
- So the probability is 2/8 = 1/4.
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