Question

In each case, are the two events dependent or independent? Toss a dime, and then toss a quarter. Pick a card from a deck of cards. Keep the card and pick another.

Answer 1

Two events are independent if the knowledge of the first does not affect the probabilities of the outcomes of the second.

The first case represents independent events. In fact, when you toss the dime, you have no further information on the result of the tossing of the quarter - it's still heads or tails with probability 50/50.

On the other hand, the second case represents dependent events. In fact, when you pick the first card, you have probability 1/52 of getting any card from the deck. Since you keep the card before picking the next one, the second picking will not have the same probability: assume the first pick was the ace of hearts. For the second pick, you are sure that the ace of hearts can't be picked, and every other card has probability 1/51 of being picked. In other words, the knowledge of the first pick - the ace of hearts - changed the probabilities of the second pick, so the events are dependent.