Question

(a)
Tim chooses one card at random from a deck of
cards, where each card ​has a single number from 1
to 13
and each number appears on
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
" and “Tim’s second ​card is a
" are not independent.

Answer 1

Two draws are without replacement, so what happens on the first draw changes the deck for the second draw.

Let A = “first card is a 7” and B = “second card is a 7” (same reasoning holds for any fixed rank). Then
- P(A) = 4/52.
- P(B | A) = after one 7 is removed there are 3 sevens left out of 51 cards, so P(B | A) = 3/51.
- P(B) (unconditional) = 4/52.

Because P(B | A) = 3/51 ≠ 4/52 = P(B), knowing that the first card was a 7 changes the probability the second is a 7. Hence A and B are not independent (equivalently P(A ∩ B) = (4/52)(3/51) ≠ (4/52)(4/52)).