Answer¶
$$= \dfrac{11^{12}}{12^{12}} = \dfrac{3138428376721}{8916100448256} \approx 35.20\%$$The plot below shows the gradual change in transmission fluid age.
From Travis Henry comes some vehicular trouble:
You want to change the transmission fluid in your old van, which holds 12 quarts of fluid. At the moment, all 12 quarts are “old.” But changing all 12 quarts at once carries a risk of transmission failure.
Instead, you decide to replace the fluid a little bit at a time. Each month, you remove one quart of old fluid, add one quart of fresh fluid and then drive the van to thoroughly mix up the fluid. (I have no idea if this is mechanically sound, but I’ll take Travis’s word on this!) Unfortunately, after precisely one year of use, what was once fresh transmission fluid officially turns “old.”
You keep up this process for many, many years. One day, immediately after replacing a quart of fluid, you decide to check your transmission. What percent of the fluid is old?
Consider that one day, many years from now.
$\dfrac{1}{12}$ of the transmission fluid is brand new.
From the previous month, $\dfrac{1}{12} \cdot \dfrac{11}{12}$ of the transmission fluid will be 1 month old.
From two months ago, $\dfrac{1}{12} \cdot \left(\dfrac{11}{12}\right)^2$ of the transmission fluid will be 2 months old.
...
From eleven months ago, $\dfrac{1}{12} \cdot \left(\dfrac{11}{12}\right)^11$ of the transmission fluid will be 11 months old.
All remaining transmission fluid is 12 months or older.
$$= 1 - \left(\dfrac{1}{12} \cdot \dfrac{1 - \left(\dfrac{11}{12}\right)^{12}}{1 - \dfrac{11}{12}}\right)$$
$$= 1 - \dfrac{12^{12} - 11^{12}}{12^{12}}$$
The plot below shows the gradual change in transmission fluid age.