If you have single pile of 120 cards, there is only 1 joker in the set of 120 cards. You then randomly split the single pile into 2 unequal piles: pile A contains 20/120 cards and pile B contains 100/120 cards. You then pick a pile and pick a single card randomly from the selected pile.
A. Which pile would most likely contain the joker?
B. What is the probability you will be able to select the correct pile and the correct card?
My answer:
A. Since the sum of all probability should equal 1 20/120 + 100/120 = 120/120
Pile B would have a higher probability.
B. P(X and Y) = P(X) * P(Y)
Picking pile A and drawing the joker: 1/20 × 20/120 = 1/120
Picking pile B and drawing the joker: 1/100 × 100/120 = 1/120
So it does not matter what pile I pick, I am just as likely to draw a random joker from each pile.
Does this make sense? If so, why? Shouldn't picking pile B always have the best probability of randomly drawing a joker the first time?
The main idea is that the joker is more likely to be in the larger pile, but you are less likely to draw it from the pile if it's in the larger pile. These two ideas cancel each other out, meaning it is irrelevant which pile you pick, as shown by your work.
Think of it like this: imagine instead of piles, you just mentally divide the deck into the top twenty cards and the bottom one hundred cards. Picking which section and then picking a card from that section is exactly the same as just picking a card from the entire deck.