The Sorry State Lottery requires you to select five different numbers from 0 thr

Question

The Sorry State Lottery requires you to select five different numbers from 0 through 68. (Order is not important.) You are a Big Winner if the five numbers you select agree with those in the drawing, and you are a Small-Fry Winner if four of your five numbers agree with those in the drawing. (Enter your answers as exact answers.)

What is the probability of being a Big Winner?


What is the probability of being a Small-Fry Winner?


What is the probability that you are either a Big Winner or a Small-Fry Winner?

Explanation / Answer

There are 69 numbers from 0-68.

a)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    69      
K = number of successes in the population =    5      
n = sample size =    5 [you choose 5]      
x = number of successes in the sample =    5 [must have 5 correct for big winner]      
          
Thus,          
          
P(   5   ) =    8.89797*10^-8 [ANSWER]

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b)

Note that the probability of x successes out of n trials is          
          
P(x) = C(N-K, n-x) C(K, x) / C(N, n)          
          
where          
N = population size =    69      
K = number of successes in the population =    5      
n = sample size =    5      
x = number of successes in the sample =    4      
          
Thus,          
          
P(   4   ) =    2.84735*10^-5 [ANSWER]

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c)

Thus,

P(4) + P(5) = 8.89797*10^-8 + 2.84735*10^-5 = 2.85625*10^-5 [ANSWER]

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