Question

(a)Tim chooses one card at random from a deck of 52 cards, where each card has a single number from 1 to 13 and each number appears on 4 different colored cards.

He holds his selected card in his hand and chooses another card at random from the remaining cards in the deck.



Part A

Explain why the events “Tim’s first card is a "9 and “Tim’s second card is a 9" are not independent.

Answer 1

Let A = "first card is a 9" and B = "second card is a 9."

With no replacement,
- P(A) = 4/52 = 1/13.
- P(B) unconditional (by symmetry) = 4/52 = 1/13.

But if A occurs, one 9 has been removed, so
- P(B | A) = 3/51 = 1/17 ≠ 1/13.

Since P(B | A) ≠ P(B), A and B are not independent. Equivalently,
P(A and B) = (4/52)(3/51) = 1/221, which is not equal to P(A)P(B) = (1/13)^2 = 1/169. The probabilities change because drawing without replacement alters the deck composition.