Consider a state lottery game in which each winning combination and each ticket consists of one set
Question:
a. Prove that if n < 2k − 1, then every winning combination has at least one pair of consecutive numbers. For the rest of the problem, assume that n ≥ 2k − 1.
b. Let i1 < . . . < ik be an arbitrary possible winning combination arranged in order from smallest to largest. For s = 1, . . . , k, let js = is − (s − 1). That is,
j1 = i1,
j2 = jk
− 1
.
.
.
jk = ik − (k − 1).
Prove that (i1, . . . , ik) contains at least one pair of consecutive numbers if and only if (j1, . . . , jk) contains repeated numbers.
c. Prove that 1≤ j1 ≤ . . . ≤ jk ≤ n − k + 1 and that the number of (j1, . . . , jk) sets with no repeats is (n−k+1/k).
d. Find the probability that there is no pair of consecutive numbers in the winning combination.
e. Find the probability of at least one pair of consecutive numbers in the winning combination.
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Related Book For
Probability And Statistics
ISBN: 9780321500465
4th Edition
Authors: Morris H. DeGroot, Mark J. Schervish
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