Airport Metal Detectors. Of people passing through an airport metal detector, 0.

Question

Airport Metal Detectors. Of people passing through an airport metal detector, 0.5% activate it (that is, p(activation)=0.005). Let X=the number among a randomly selected group of 500 people that activate the detector. Note that X~ Bin(500,0.005).

a. Calculate the probability that five of the people activate the detector (to 6 digits) using the Binomial pmf.

b. Can the Poisson approximation to the Binomial be used here? Check the conditions.

c. Calculate the approximate probability that five of the people activate the detector based on the Poisson pmf (to 6 digits).

d. How much did the probabilities differ between the Binomial and the Poisson calculations?

Explanation / Answer

a)

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 =    500      
p = the probability of a success =    0.005      
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.066716262 [ANSWER]

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b)

Yes, as n = 500 > 100, p = 0.005 < 0.05.

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c)

Note that the probability of x successes is          
          
P(x) = u^x e^(-u) / x!          
          
where          
          
u = the mean number of successes =    2.5      
          
x = the number of successes =    5      
          
Thus, the probability is          
          
P (    5   ) =    0.066800943 [ANSWER]

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d)

The difference is

difference = P(binom) - P(Poi)

= 0.066716262 - 0.066800943

= -8.4681*10^-5 [ANSWER]

which is quite small.

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