# Stats help!

#### sandra2223

##### New member
A manufacturer claims that their manufacturing process makes widgets with at most 5 percent defective rate. assume that the defective rate is 5%. Let x denote the number of defective widgets in a box of 50 widgets.

What is the probabilty of finding more than 2 defective widgets in the box?

what is the variance of X?

#### soroban

##### Elite Member
Hello, Sandra!

The first part is a binomial probability.

A manufacturer claims that their manufacturing process makes widgets with at most 5% defective rate.
Let x denote the number of defective widgets in a box of 50 widgets.

What is the probabilty of finding more than 2 defective widgets in the box?

The opposite of "more than two" is "two or less".

. . $$\displaystyle \begin{array}{ccccccc}P(\text{0 def}) &=& (0.95)^{50} &=& 0.076944975 \\ P(\text{1 def}) &=& {50\choose1}(0.05)(0.95)^{49} &=& 0.202486777 \\ \\[-3mm] P(\text{2 def}) &=& {50\choose2}(0.05)^2(0.95)^{48} &=& 0.261101370 \\ \hline & &\text{Total:} && 0.540533122\end{array}$$

$$\displaystyle \text{Hence: }\(\text{2 or less def}) \:\approx\:0.54 \:=\:54\%$$

$$\displaystyle \text{Therefore: }\(\text{more than 2 def}) \;=\;100\% - 54\% \;=\;46\%$$

#### sandra2223

##### New member
Thank you so much. I really appreciate it. I still need help with finding the variance of X, i cant figure it out.