# Find probability

Let $X$ (millimeters) be the thickness of the washer. Assume that $X$ has density

$$f(x)= \begin{cases} kx & \text{if 0.9 \leq x \leq 1.1};\\ 0 & \text{otherwise}.\end{cases}$$

What is the probability that a washer has thickness between $0.95\text{ mm}$ and $1.5\text{ mm}$

here is what I got. First I find $k$ and got $k=5$.

Then I find $F(x)$ and got $F(x)= \dfrac{5x^2}{2} -2.025$.

$$P(0.95\leq x\leq 1.5)= F(1.5)-F(0.95)=1-\dfrac{5(0.95)^2}{2} +2.025=76.87\%$$

However, the book say it should be $50\%$. Did I do something wrong?

• I believe your answer is correct. I integrated the pdf from 0.95 to 1.5 and got 0.76875
– ved
Jul 2, 2014 at 15:41
• "Did I do something wrong?" Yes, you miscopied "between 0.95 mm and 1.05 mm" and replaced it by "between 0.95 mm and 1.5 mm".
– Did
Jul 2, 2014 at 19:36

I think there is a typo:

$\qquad$What is the probability that a washer has thickness between $0.95\text{ mm}$ and $1.5\text{ mm}$

should be:

$\qquad$What is the probability that a washer has thickness between $0.95\text{ mm}$ and $1.05\text{ mm}$.

Then,

$$P(0.95\leq x\leq 1.05)= F(1.05)-F(0.95)=\dfrac{5(1.05)^2}{2} +2.025-\dfrac{5(0.95)^2}{2} +2.025=50\%$$.

I got the same result as you.

I think it cannot be $50\%$ because the density is increasing by a factor $k=5>1$, which means the probability above the middle $x=1$ is larger than $50\%$. So the probability including below the middle $0.95$ cannot be $50\%$.