Discrete Math HELP!?

Suppose that you buy a lottery ticket containing k distinct numbers from among 

{1, 2, . . . , n} where 1 ≤ k ≤ n. To determine the winning tickets, k balls are randomly drawn without replacement from a bin containing n balls numbered 1, 2, . . . , n. 

(a) What is the probability that at least one of the numbers on your lottery ticket is among those drawn from the bin?(b) Prizes start when you have at least (k /2) (ie. half) numbers in common with the drawn balls. Express the probability that you win something. You may find it helpful to use summation notation.

*I did question 1 but I am unaware on how to do question 2. I don't get what the question is asking me. Can someone please help me out? Thanks.

1 Answer

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  • Alan
    Lv 7
    1 month ago

    (b)  part b uses  the hyper-geometric probability distribution

    let j = number of  matches  

    P(j > = K/2  )  = P( j=floor( ((k/2) + 0.5)  ) + P( j= floor ((k/2) + 1.5)  ) 

    + P( j=   floor((k/2) + 2.5) + ... + P(j= (k-1) )  + P(j= k) 

    To find each probability  you can treat that as hyper-geometric probability 

    So in general ,pick exactly k items in subset of size n chosen 

    from an original Group size N where there were K of the 

    particular item in the larger group

    This cases can be solved with the hypergeometric

     distribution 

    https://en.wikipedia.org/wiki/Hypergeometric_distr...

    P(X =k) = ( ( K over k )( (N-K) over (n-k)) /  ( N over n )

    ===where "over" means combination 

    or in terms of combinations 

    P(X=k) = (C(K,k)*C( (N-K), n-k)/ ( C(N,n)   

    But in this case what does N,n,K, k   

    equal 

    N = n 

    n = k  

    K = k  

    k = j  and 

    j increments as indicated above  

    so putting in the formula  

    P(X = j ) = ( C(k, j) * C( n-k, k-j) / (C(n,k)    

    P(X = j)  =C(k,j) * C(n-k, k -j ) / C(n,k)

    P( you win ) = from j = floor ( (k/2) + 0.5)  to  k    

    Σ C(k,j)* C(n--k,k-j)/C(n,k) 

    Example  N = 20 

    k = 6  f

    loor( 6/2 + 1/2) = 3 

    P( you win ) = from 3 to 6  Σ  C(6,j)*C(14,6-j)/ C (20,6)   

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