A lottery involves picking 6 numbers from the numbers 1 to 5

A lottery involves picking 6 numbers from the numbers 1 to 51. The probability of picking 6 winning numbers is 1/18,009,460. Supose 10 million individuals participate during a week. Assuming independence, find the,

a) Probability that there is no winner

b) probability that there is at least one winner

c) expected number of winners

Solution

A)

Note that the probability of x successes out of n trials is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes = 10000000*(1/18009460) =    0.555263734      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.573920879 [ANSWER]

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b)

P(At least one) = 1 - P(0) = 0.426079121 [ANSWER]

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c)

E(x) = n p = 10000000*(1/18009460) = 0.555263734 [ANSWER]