The lottery commission announces they will print 100,000 lottery tickets, of whi

Question

The lottery commission announces they will print 100,000 lottery tickets, of which 120 will be jackpotwinning tickets. You are thinking of randomly purchasing 200 tickets, but first want to find the probability that at least one of the 200 will be a jackpot-winning ticket. You immediately recognize this as a Hypergeometric problem. But being clever, you notice that n is significantly smaller than N, so you decide to simplify your life a little and use the Binomial distribution as an approximation of the Hypergeometric distribution. But then you also notice that since p is small and n is large, you can simplify things further by using the Poisson distribution as an approximation of the Binomial distribution. Using the Poisson distribution as an approximation of the Binomial distribution, where the Binomial distribution is being used as an approximation of the Hypergeometric distribution, find the probability at least one of your tickets is a winning ticket.

Explanation / Answer

The binomial probability of winning is then

p = x/N = 120/100000 = 0.0012

hence, out of 200 tickets, the mean number of winning tickets is

E(x) = n p = 200*0.0012 = 0.24

Note that P(at least 1) = 1 - P(0).

Note that the probability of x successes is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    0.24      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.786627861

Thus, P(at least 1) = 1 - P(0) = 0.213372139 [ANSWER]

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