There's no definitive answer since the text doesn't give any probability law that would follow the openings of accounts.
Example 1 : suppose every day, exactly 10 new accounts are opened (then the average is obviously 10).
Then there's 100% chance for more than 8 accounts being opened today (since 10 will certainly be opened).
Example 2 : suppose the opening of accounts follows a Poisson's law of parametre 10 (hence, an average of 10 accounts opened every day). This is what is usually chosen for such modelisation.
"initiations qualify as rare events" indicates the author of the text wants you to apply Poisson's law (even if it's not written...)
Let X the number of accounts opened on a given day.
For every natural number k, P (X=k) = 10^k * exp(-10) / k!
(i) We're looking for P (X > 8).
P (X > 8) = 1 - P (X < 9)
And P(X < 9) = P(X=0) + P(X=1) + ... + P(X=8)
P(X < 9) = exp(-10) * ( 10^0 / 0! + 10^1 / 1! + ... + 10^8 / 8! )
~ 0.33282
Finally, P (X > 8) ~ 1 - 0.33282 ~ 0.66718 ~ 66.7 %
(ii) If Y is the number of new accounts the day after, we're looking for P(X+Y>16).
P(X+Y>16) = 1 - P(X+Y<17)
Now suppose X and Y are independant (which is not said in the text). Then X+Y follows a Poisson's law of parametre 20 (=10+10).
Then :
P(X+Y<17) = P(X+Y=0) + P(X+Y=1) + ... + P(X+Y=16)
= exp(-20) * (20^0 / 0! + 20^1 / 1! + ... + 20^16 / 16! )
~ 0.221
Finally, P(X+Y>16) ~ 1 - 0.221 ~ 0.779 = 77.9%
Notice that "more than 8" means 9 or more, and "more than 16" means 17 or more.
I'm not sure if this phrasing was intentional : there's nothing to compare between 9+ and 17+, whereas you could say something interesting when comparing 8+ and 16+.
The text should have asked in this case : "more than 7" and "more than 15".
Maybe your solution was refering to this ?
Good luck !