Compare the Poisson approximation with the correct binomial

Compare the Poisson approximation with the correct binomial probability for the following cases: If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1/100, what is the (approximate) probability that you will win a prize

Solution

a)

Binomial:

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    8      
p = the probability of a success =    0.1      
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.14880348 [ANSWER, BINOMIAL]

....

POISSON:

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 =    0.8      
          
x = the number of successes =    2      
          
Thus, the probability is          
          
P (    2   ) =    0.143785269 [ANSWER, POISSON]

They are quite close here.

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b)

BINOMIAL:

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.95      
x = the number of successes =    9      
          
Thus, the probability is          
          
P (    9   ) =    0.315124705 [ANSWER, BINOMIAL]

POISSON:

Here, instead of counting successes, we count \"failures\" as successes, so that n = 10, p = 0.05, x = 1.

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 =    0.5      
          
x = the number of successes =    1      
          
Thus, the probability is          
          
P (    1   ) =    0.30326533 [ANSWER, POISSON]

They are close in this one.

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c)

BINOMIAL:
Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    10      
p = the probability of a success =    0.1      
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.34867844 [ANSWR, BINOMIAL]

POISSON:

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 =    1      
          
x = the number of successes =    0      
          
Thus, the probability is          
          
P (    0   ) =    0.367879441 [ANSWER, POISSON]

They are not very close here, but maybe close enough for some purposes.

******************

D)

BINOMIAL:

Note that the probability of x successes out of n trials is          
          
P(n, x) = nCx p^x (1 - p)^(n - x)          
          
where          
          
n = number of trials =    9      
p = the probability of a success =    0.2      
x = the number of successes =    4      
          
Thus, the probability is          
          
P (    4   ) =    0.066060288 [ANSWER, BINOMIAL]

POISSON:

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 =    1.8      
          
x = the number of successes =    4      
          
Thus, the probability is          
          
P (    4   ) =    0.072301734 [ANSWER, POISSON]

Here, they are also close.

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