A brand name has a 60% recognize rate. Assume the owner of the brand wants to ve

Question

A brand name has a 60% recognize rate. Assume the owner of the brand wants to verify that rate by beginning with a small sample of 8 randomly selected consumers.

a) the probability that exactly 7 of the 8 selected consumers recognize the brand name?

b) What is the Probability that all selected consumers recognize the brand name?

c) What is the probabililty that at least 7 of the selected consumers recognize the brand name?

Explanation / Answer

Binomial 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

a.

P( X = 7 ) = ( 8 7 ) * ( 0.6^7) * ( 1 - 0.6 )^1
= 0.0896

b.
P( X = 8 ) = ( 8 8 ) * ( 0.6^8) * ( 1 - 0.6 )^0
= 0.0168
c.
P( X < 7) = P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 8 6 ) * 0.6^6 * ( 1- 0.6 ) ^2 + ( 8 5 ) * 0.6^5 * ( 1- 0.6 ) ^3 + ( 8 4 ) * 0.6^4 * ( 1- 0.6 ) ^4 + ( 8 3 ) * 0.6^3 * ( 1- 0.6 ) ^5 + ( 8 2 ) * 0.6^2 * ( 1- 0.6 ) ^6 + ( 8 1 ) * 0.6^1 * ( 1- 0.6 ) ^7 + ( 8 0 ) * 0.6^0 * ( 1- 0.6 ) ^8
= 0.8936
P( X > = 7 ) = 1 - P( X < 7) = 0.1064

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