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Copyright © 19962018 Alexander Bogomolny
Let r_{n}(p) be the sought expectation. We shall show that
(1)  r_{n}(p) = p + (n  1)p(1  p) 
by induction on n.
When n = 1, (1) becomes r_{1}(p) = p, which is clearly true.
Suppose n > 1 and assume (1) holds for r_{n1}(p), which is the expected number of ranges for an

References
 G. G. Chappell, A Quicky, Math Magazine, Vol. 72, No. 4, October 1999, Q893 (p. 326), A893 (p. 331).
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Copyright © 19962018 Alexander Bogomolny