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A Dicey Paradox

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Nines and Tens

Explain why it is that when you throw two dice you are more likely to get a score of 9 than of 10. What about the case of 3 dice? Is a score of 9 more likely then a score of 10 with 3 dice?

Tying the Rope

Stage: 3 Short Challenge Level: Challenge Level:1


There are six different possible ways of choosing two of the loose ends:

Imagine calling the ends of one rope A and B, and the ends of the other rope X and Y.

These are all the different possible pairings:
A & B
A & X
A & Y
B & X
B & Y
X & Y

Four of these pairings lead to the ropes being tied together:

A & X
A & Y
B & X
B & Y

Therefore the probability that Sam is left holding one piece of rope is $\frac{4}{6}=\frac{2}{3}$.

Alternative method:

When Pat picks up one end, there are three possible ends left that she can choose. One of these will be the other end of the same rope and will produce a loop. Either of the other two will produce the desired result.

Therefore the probability that Sam is left holding one piece of rope is $\frac{2}{3}$.







This problem is taken from the UKMT Mathematical Challenges.