Answer:
The probability that more than a third of the buyers would prefer brown is 0.8491.
Step-by-step explanation:
This is binomial problem with n=15 and p=0.50.
The binomial formula of probaility is:
P(X=x)=[tex]\frac{n!}{x!(n-x)!} p^{x} (1-p)^{n-x}[/tex]
More than a third of the buyers: 15÷3=5
We have to find P(x>5).
P(x>5) = 1 - P(x≤5) = 1 - P(x=0) - P(x=1) - P(x=2) - P(x=3) - P(x=4) - P(x=5)
P(x=0)=[tex]\frac{15!}{0!15!} (0.5)^{0} (1-0.5)^{15}[/tex]=0.00003
P(x=1)=[tex]\frac{15!}{1!14!} (0.5)^{1} (1-0.5)^{14}[/tex]=0.00046
P(x=2)=[tex]\frac{15!}{2!13!} (0.5)^{2} (1-0.5)^{13}[/tex]=0.00320
P(x=3)=[tex]\frac{15!}{3!12!} (0.5)^{3} (1-0.5)^{12}[/tex]=0.01389
P(x=4)=[tex]\frac{15!}{4!11!} (0.5)^{4} (1-0.5)^{11}[/tex]=0.04165
P(x=5)=[tex]\frac{15!}{5!10!} (0.5)^{5} (1-0.5)^{10}[/tex]=0.09164
P(x>5) = 1 - P(x≤5)= 1- 0.00003 - 0.00046 - 0.00320 - 0.01389 - 0.04165 - 0.09164 = 0.8491
