Answer to Question #264204 in Statistics and Probability for Honour

Question

the manufacturer of a bag of sweets clams that there is a 90% chance that the bag contains some toffees. if 20 bags are chosen, what is the probability that 

a) All the bags contain toffees   

b) More than 18 bags contain toffees 

 

Answer 1

Let $X=$ the number of the bags which contain some toffees: $X\sim Bin(n, p).$

Given $n=20, p=0.9, q=1-p=1-0.9=0.1.$

a)

$$P(X=20)=\dbinom{20}{20}(0.9)^{20}(0.1)^{20-20}$$

$$\approx0.1215766546$$

b)

$$P(X>18)=P(X=19)+P(X=20)$$

$$=\dbinom{20}{19}(0.9)^{19}(0.1)^{20-19}+\dbinom{20}{20}(0.9)^{20}(0.1)^{20-20}$$

$$\approx0.2701703435+0.1215766546$$

$$\approx0.391747$$