For the bowl with 2 poisonous apples out of 5, and you have to eat 3, there are 3 possible ways to die:
The first way to die is by eating 2 poisonous apples in a row. The first time there's a 2/5 chance to eat a poisonous apple, which reduces to a 1/4 chance to eat a second one. Multiply these two together:
(2/5)*(1/4) = 10%
Second way to die is if you eat one good apple and then two bad ones:
(2/5) * (2/4) * (1/3) = 6.6%
Third way to die is if you eat one bad one, then a good one, and then a bad one:
(2/5) * (1/4) * (1/3) = 3.33%
Add these probabilities together,
10% + 6.6% + 3.33% = (Basically) 20%
This means that the chance of dying is 20%, so a 80% chance of survival.
For the second bowl there is only one way to die, and that's if you eat 2 bad ones in a row:
(3/5) * (2/4) = 30%
So that means there's a 30% chance of death and a 70% chance of survival, respectively.
I can't figure out how people get to it being 70% for both.