Respuesta :
Solution :
It is given that about 37% of the adults say cashew nuts are their favorite nut. And a sample of 12 adults are taken to name their favorite nut.
We note that probability of [tex]$x$[/tex] successes out of [tex]$n$[/tex] trial is given by :
[tex]$P(n,x)=^nC_x p^x (1-p)^{(n-x)}$[/tex]
Here, number of trails, n = 12
probability of success, p = 0.37
number of successes, x = 3
a). Therefore the probability of the adults to say cashew nut as their favorite of exactly three is given by :
[tex]$P(3)=^{12}C_3 (0.37)^3 (1-0.37)^{(12-3)}$[/tex]
= 0.174217909
b). We know that :
P(at least x) = 1 - P(at most x - 1)
So we use the cumulative binomial distribution table.
i.e. [tex]$P(x \geq 4) = 1 -P(x \leq3)$[/tex]
[tex]$= -1[P(x=0)+P(x=1)+P(x=2)+P(x=3)]$[/tex]
[tex]$= 1-[^{12}C_0 (0.37)^0 (0.63)^{12}+^{12}C_1 (0.37)^1(0.63)^{11}+^{12}C_2 (0.37)^2(0.63)^{10}+^{12}C_3(0.37)^3(0.63)^9]$[/tex]= 0.70533
Therefore, P(at least 4) = 0.70533
c). [tex]$P(x \leq 2) = P(x=0) + P(x=1)+P(x=2)$[/tex]
[tex]$=^{12}C_0(0.37)^0(0.63)^{12}+^{12}C_1(0.37)^1(0.63)^{11}+^{12}C_2(0.37)^2(0.63)^{10}$[/tex]
= 0.12045205
Therefore, P(at most 2) = 0.12045205
