Answer to Question #151816 in Statistics and Probability for Marc Griffin

Question

In a shipment of bananas, 10% are rejected because they are too small. What is the probability that in a box of 80 bananas at most 8 will be rejected because they are too small?

Answer 1

Here

$n=80$

$p=10\%=0.1$

Here x follows the binomial distribution:

$$P(X=x)=C(n,x)\cdot p^x\cdot (1-p)^{n-x}$$

$P(X\le8)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)+P(X=7)+P(X=8)$ $P(X=0)=C(80,0)\cdot 0.1^0\cdot (1-0.1)^{80-0}=0.00022$

$P(X=1)=C(80,1)\cdot 0.1^1\cdot 0.9^{79}=0.00194$$P(X=2)=C(80,2)\cdot 0.1^2\cdot 0.9^{78}=0.00852$

$P(X=3)=C(80,3)\cdot 0.1^3\cdot 0.9^{77}=0.02462$

$P(X=4)=C(80,4)\cdot 0.1^4\cdot 0.9^{76}=0.05266$

$P(X=5)=C(80,5)\cdot 0.1^5\cdot 0.9^{75}=0.08895$

$P(X=6)=C(80,6)\cdot 0.1^6\cdot 0.9^{74}=0.12354$

$P(X=7)=C(80,7)\cdot 0.1^7\cdot 0.9^{73}=0.14510$

$P(X=8)=C(80,8)\cdot 0.1^8\cdot 0.9^{42}=0.14712$

$P(X\le8)=0.00022+0.00194+0.00852+0.02462+0.05266+0.08895+0.12354+0.14510+0.14712=0.59267$

Answer: the probability that in a box of 80 bananas at most 8 will be rejected because they are too small is 0.59267.