1/2 of max HP, no status

You have a 10.661% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 7 balls and at least a 95% chance of catching it within 27 balls.

1/2 of max HP, paralyzed

You have a 14.34% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 5 balls and at least a 95% chance of catching it within 20 balls.

1/2 of max HP, asleep

You have a 19.752% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 4 balls and at least a 95% chance of catching it within 14 balls.

1 HP, no status

You have a 14.34% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 5 balls and at least a 95% chance of catching it within 20 balls.

1 HP, paralyzed

You have a 23.416% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 3 balls and at least a 95% chance of catching it within 12 balls.

1 HP, asleep

You have a 33.695% chance of capturing it per ball. Thus, you have at least a 50% chance of catching it within 2 balls and at least a 95% chance of catching it within 8 balls.

source

The calculator unfortunately doesn't tell me how many you need for 70% catch chance, so I decided to figure it out. When you flip a coin once, you have a 1/2 chance of heads and a 1/2 chance of tails. When you flip it twice, the possible results are TT, TH, HT, and HH. So there's a 1/4 chance of getting 2 tails in a row and a 3/4 chance of getting at least one heads.

With 3 flips, it's TTT, TTH, THT, THH, HTT, HTH, HHT, and HHH, with a 1/8 chance of 3 tails in a row and 7/8 chance of getting at least one heads.

With n flips, there's a (1/2)^n chance of getting n tails in a row, and a 1 - (1/2)^n chance of getting at least one heads. Let's say you want at least a 70% chance of getting at least one heads.

1 - (1/2)^n = 70% = 7/10

1 - (1/2)^n - 1 = 7/10 - 1

-(1/2)^n = -3/10

(1/2)^n = 3/10

log(1/2, (1/2)^n) = log(1/2, 3/10)

n = log(1/2, 3/10) coins, which is close to 1.7 and rounds up to 2 coins because you can't flip 0.7 coins.

If the chance of heads is p (which can be 1/2 or something else), the chance of tails is 1 - p. There's a (1 - p)^n chance of getting n tails in a row, and a 1 - (1 - p)^n chance of getting at least one heads. This is the chance of getting at least one heads.

1 - (1 - p)^n = 70% = 7/10

1 - (1 - p)^n - 1 = 7/10 - 1

-(1 - p)^n = -3/10

(1 - p)^n = 3/10

log(1 - p, (1 - p)^n) = log(1 - p, 3/10)

n = log(1 - p, 3/10)

I'll use n = ceiling(log(1 - p, 3/10)) because you can't flip a fraction of a coin or throw a fraction of a great ball.

(these aren't real math proofs, I just tried and failed to write an explanation that makes sense)

Now we can take the p values from earlier and put them into the formula.

1/2 of max HP, no status: n = ceiling(log(1 - 0.10661, 3/10)) = 11 great balls

1/2 of max HP, paralyzed: n = ceiling(log(1 - 0.1434, 3/10)) = 8 great balls

1/2 of max HP, asleep: n = ceiling(log(1 - 0.19752, 3/10)) = 6 great balls

1 HP, no status: n = ceiling(log(1 - 0.1434, 3/10)) = 8 great balls

1 HP, paralyzed: n = ceiling(log(1 - 0.23416, 3/10)) = 5 great balls

1 HP, asleep: n = ceiling(log(1 - 0.33695, 3/10)) = 3 great balls