Question

Tim chooses one card at random from a deck of 52 cards, where each card has s 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.
explain why the events " tims first card is a 9" and "tims second is a 9" are not independent.

Answer 1

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

P(A) = 4/52 = 1/13.

If the first card is a 9, there are only 3 nines left among 51 cards, so
P(B | A) = 3/51 = 1/17.

Since P(B | A) ≠ P(B) (and in fact P(B) = 4/52 = 1/13 by symmetry), the events are not independent. Equivalently,
P(A ∩ B) = (4/52)(3/51) = 1/221 ≠ (1/13)(1/13) = 1/169 = P(A)P(B).

Intuitively: because cards are drawn without replacement, seeing a 9 first changes how many 9s remain for the second draw, so the outcome of the first draw affects the probability of the second.