Three brothers in a random split

Topic: Random experiments and probability · more questions on this topic
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Twenty people include three siblings. The twenty are split at random into two unlabelled groups of ten. Find the probability that all three siblings land in the same group.

Worked answer (click to show)

Choose 10 of the 20 people for one group; the rest form the other. There are \binom{20}{10} ways to do this. For all three siblings to sit in the chosen group, the other 7 members are chosen from the other 17 people, in \binom{17}{7} ways. The same count applies to the other group. The two groups are labelled by the choice of the first group in this counting, so

\mathbb{P}=\frac{2\binom{17}{7}}{\binom{20}{10}}=\frac{4}{19}\approx 0.211.