After failing this quest several times, it occurred to me that I should figure out whether it was actually possible before I tried to complete it. So I did some math.
Short version: This quest, though theoretically possible, is not possible to complete in real life.
* Although we are not given actual fail rate percentages for any but the last type of fish, it seems reasonable to assume that the fish show a progression like 10% fail rate, 20%, 30%, 40%, 50%. However, I tend to fail on the second type of fish, so I think these may be a bit high. For my calculations, I use: 5%, 10%, 20%, 25%, 50%. I believe that these are conservative, which means that the numbers in reality are probably much worse than the numbers you'll see below.
* I have not gotten past the third tier of fish, so I don't actually know, but I'm assuming it's 10 fish for each tier up to tier 5.
* There will be rounding errors here because there's no reason to be overly precise when we're making up numbers in the first place.
Round 1 (estimated: 95% success chance per easy fish)
Chance of catching 10 easy fish in a row = .95^10 = approximately 60%
Note: This seems low based on my own experience of failing on this round once out of about 10 tries. However, the sample size is too low to be sure.
Round 2 (estimated: 90% success chance per simple fish)
Chance of catching 10 simple fish in a row = .90^10 = approximately 35%
Note: This seems to match my experience where I have only gotten past this round once out of about 8 or 9 tries.
Cumulative chance of success: .60 * .35 = approximately 21%
Round 3 (estimated: 80% success chance per possible fish)
Chance of catching 10 possible fish in a row = .80^10 = approximately 11%
Note: I have only gotten this far one time, so I don't have a sense of how accurate this is.
Cumulative chance of success: .60 * .35 * .11 = approximately 2%
Round 4 (estimated: 75% success chance per tricky fish)
Chance of catching 10 possible fish in a row = .75^10 = approximately 6%
Note: It seems like a big jump from 25% failure rate to 50% failure rate, so this is probably a much worse chance in reality.
Cumulative chance of success: .60 * .35 * .11 * .02 = approximately .04%
Round 5 (stated in-game: 50% success chance per unlikely fish)
Chance of catching 10 unlikely fish in a row = .50 ^ 10 = approximately .1%
Note: The game says there's an equal chance to catch a bad fish or an unlikely fish, so it is 50%. However nobody has gotten far enough to confirm that it requires 10 (at least, nobody that I'm aware of)
Cumulative chance of success: .60 * .35 * .11 * .02 * .001 = approximately .00005%
To put that in perspective:
.00005% in math terms is .0000005 = 5/10,000,000 = 1 winner in every 2 million attempts.
This year, you will get about 20 opportunities to try to win this. If you try every time it's available, you can expect to win once in the next 100,000 years.
From another perspective, you are about 4 times as likely to win $10,000 from buying one powerball lottery ticket as you are to complete this quest.
From yet another perspective, you are 10 times as likely to get killed by an asteroid falling from outer space and landing on your head as you are to complete this quest. (source: http://www.livescience.com/3780-odds-dying.html )
So, will you be trying this quest again? WHAT IF YOU GET LUCKY?!?!?!?!?!?!?