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

Two cards are drawn without replacement, so the first draw affects what remains for the second.

P(second is 9 | first is 9) = 3/51 (three 9s left in 51 cards) = 1/17 ≈ 0.0588.
But P(second is 9) = 4/52 = 1/13 ≈ 0.0769.

Because P(second is 9 | first is 9) ≠ P(second is 9), the events are not independent. (Equivalently, P(first and second are 9) = (4/52)(3/51) ≠ (4/52)(4/52) = P(first is 9)P(second is 9).)