In: Statistics and Probability
A survey found that 68% of adults ages 18 to 25 think that their generation is unique and distinct. a) If we randomly select 12 adults ages 18 to 25, what is the probability that exactly 4 of them think that their generation is not unique and distinct? b) If we randomly select 16 adults ages 18 to 25, what is the probability that more than 14 of them think their generation is unique and distinct? c) If we randomly select 20 adults ages 18 to 25, what is the probability that at least 2 of them think their generation is not unique and distinct?
Solution :
Given that P(unique and distinct) = 0.68
=> P(not unique and distinct) = 1 - P(unique and distinct) = 0.32
=> For binomial distribution , P(x = r) = nCr*p^r*q^(n-r)
a)
=> The probability that exactly 4 of them think that their
generation is not unique and distinct is 0.2373
=> from given information , n = 12 , p = 0.32 , q = 0.68
=> P(x = 4) = 12C4*0.32^4*0.68^8
= 0.2373
b)
=> The probability that more than 14 of them think their
generation is unique and distinct is
=> from given information , n = 16 , p = 0.68 , q = 0.32
=> P(x > 14) = P(x = 15) + P(x = 16)
= 16C15*0.68^15*0.32^1 + 16C16*0.68^16*0.32^0
= 0.0157 + 0.0021
= 0.0178
c)
=> The probability that at least 2 of them think that their
generation is not unique and distinct is 0.9954
=> from given information , n = 20 , p = 0.32 , q = 0.68
=> P(x >= 2) = 1 - P(x < 2)
= 1 - [P(x = 1) + P(x = 0)]
= 1 - [20C1*0.32^1*0.68^19 + 20C0*0.32^0*0.68^20]
= 1 - [0.0042 + 0.0004]
= 0.9954