5.) The television show Degenerate Housewives has been successful for many years

Question

5.) The television show Degenerate Housewives has been successful for many years. That show recently had a share of 18, meaning that among the TV sets in use, 18% were tuned to Degenerate Housewives. Assume that an advertiser wants to verify that 18% share value by conducting its own survey, and a pilot survey begins with 12 households have TV sets in use at the time of a Degenerate Housewivesbroadcast.

Find the probability that none of the households are tuned to Degenerate Housewives. 
P(none) = 

Find the probability that at least one household is tuned to Degenerate Housewives. 
P(at least one) = 

Find the probability that at most one household is tuned to Degenerate Housewives. 
P(at most one) = 

If at most one household is tuned to Degenerate Housewives, does it appear that the 18% share value is wrong? (Hint: Is the occurrence of at most one household tuned to Degenerate Housewives unusual?) 

Explanation / Answer

BIONOMIAL DISTRIBUTION
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 12 * 0.18
= 2.16
II.
variance = npq
where
n = total number of repetitions experiment is excueted
p = success probability
q = failure probability
variance = 12 * 0.18 * 0.82
= 1.7712
III.
standard deviation = sqrt( variance ) = sqrt(1.7712)
=1.33086
a.
P( X = 0 ) = ( 12 0 ) * ( 0.18^0) * ( 1 - 0.18 )^12
= 0.0924
b.
P( X < 1) = P(X=0)
= ( 12 0 ) * 0.18^0 * ( 1- 0.18 ) ^12
= 0.0924
P( X > = 1 ) = 1 - P( X < 1) = 0.9076
c.
P( X < = 1) = P(X=1) + P(X=0)
= ( 12 1 ) * 0.18^1 * ( 1- 0.18 ) ^11 + ( 12 0 ) * 0.18^0 * ( 1- 0.18 ) ^12
= 0.3359, no it is not unusual event

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