I think what you did was multiply the two probabilities together as if they were independent events. That doesn't seem right to me. Something like flipping a coin twice would be two independent events. Whichever result the first coin toss is, heads or tails, has no effect on the outcome of the second flip.
In @savvyplayer's problem the first day has a 50% chance. Either the user changes the profile picture or the user doesn't. If the user does change the picture the first day than that event effects the second day event's probability by negating the possibility of a 60% chance of changing the picture within both days.
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No, what I did was consider everything as an independent event. I didn't feel like thinking too much. Is it possible for someone to change on day 1 and 2? Yes but I am not counting it.
Day 1 = 50%
Day 1+2 = 60%
60%-50%=10% = % on Day 2.
O.K. Now you've shown your work so nobody else needs to try reverse engineering what you might have done to arrive at your answer. 😀