The answers for this one are given below, but I really dont understand how to ar
ID: 3338305 • Letter: T
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The answers for this one are given below, but I really dont understand how to arrive at them. If you can please show an explanation and steps that'd be great, thanks!
(23) Review The State Lottery chooses r numbers from a list of the first n positive integers {1,2,3,...,n) (without replacement). Consider the event this happens to be S. What is the probability that:1 (a) S contains no consecutive integers? (b) S includes exactly one pair of consecutive integers? (c) the numbers in S are drawn in increasing order? .13 (e) when you play this lottery exactly k of your choices matches S?Explanation / Answer
(a) Firstly r integers can be chosen out of n in nCr ways.
Let us consider the integers immediately AFTER the r integers we will choose eventually. These are forbidden numbers.
There are r integers except if we choose the nth number. If we do choose the nth number, we have n-1 other numbers to choose and r forbidden numbers (r-1 + the nth number iself.)
Total number of numbers we can choose = (n-r)Cr + (n-r)C(r-1) which by identity equals (n-r+1)Cr.
Probability = (n-r+1)Cr / nCr.
(b) Since there is one pair of consecutive integers, let us call this pair of integers as a BIG integer. Note that there are r - 1 choices for the pair. We thus have to choose r-1 integers (small or big) among n - 1 integers (small or big).
The forbidden numbers in this case are r - 1.
By using earlier logic this can be done with probability
(r - 1) (n - 1 - (r - 1))C(r - 1) / nCr = (r-1) (n-r+1)C(r-1) / nCr
(c) There are r! ways to arrange the r numbers we choose.
Among them one is ascending order.
The probability is 1/r!.
(d) As noted earlier there are nCr ways to choose r numbers.
Thus the choice of numbers being same as S is 1/nCr.
(e) The k choices can be chosen out of r in rCk ways.
The n-r numbers that do not match have r - k choices.
The probability is rCk (n-r)C(r-k) / nCr.
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