A common bit of financial advice you may have heard is "time in the market beats timing the market." As with many bits of advice coming in bite$-$size, catchphrase form, they are worth examining more carefully.

Say you have $$10,000$ to invest. The S&P $500$ over the past $70$ years has had an average annual return of $10\%.$ Rather, and investment of $$1\mathrm{.}00$ would net a return of $$0\mathrm{.}10$ totalling to $$1\mathrm{.}10.$ "Time in the market" is meant to rely and expect this average rate of return over an extended period of time, while "timing the market" is meant as investing just before a spike in rates. The latter would lead to a more immediate jump in return while still having access to the investment amount beforehand.

The issue comes to how do you know when to invest?

$($a$)$ Say you invest $$10,000$ today. A friend waits $10$ years, investing $$10k$ themselves in $2034.$ In $2044,$ you experienced an average return of $10\%$ while your friend experience twice that, with an average return of $20\%.$ How much do each of you have in $2044?$

$($b$)$ Interpret your result from $($a$).$ What does this say about the aforementioned advice? Do you agree$/$disagree with it$,$ personally?