1.12 Supplementary Exercises - Probability and Statistics

Exercise 1.12.13¶

Consider a state lottery game in which each winning combination and each ticket consists of one set of $k$ numbers chosen from the numbers 1 to $n$ without replacement. We shall compute the probability that the winning combination contains at least one pair of consecutive numbers.

(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 $n \geq 2k − 1$ .
(b) Let $i_1 < \cdots < i_k$ be an arbitrary possible winning combination arranged in order from smallest to largest. For $s = 1, \ldots, k$ , let $j_s = i_s - (s-1)$ . That is,
$\begin{align*} j_1 &= i_1, \\ j_2 &= i_2 - 1 \\ &\vdots \\ j_k &= i_k - (k-1). \end{align*}$
(1)
Prove that $(i_1, \ldots, i_k)$ contains at least one pair of consecutive numbers if and only if $(j_1, \ldots, j_k)$ contains repeated numbers.
(c) Prove that $1 \leq j_1 \leq \cdots \leq j_k \leq n - k + 1$ and that the number of $(j_1, \ldots, j_k)$ sets with no repeats is $\binom{n-k+1}{k}$ .
(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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