Probability and independent or dependent events

Question

A standard deck of playing cards contains 52 total cards, 13 of which are "spades".

Suppose you and a friend are playing a game. You can choose any two cards from the deck without looking. If the two cards you choose are both spades, you win!

You pull out one card, and then another, without putting the first one back.

What is the approximate probability that you win?

Are your two choices independent or dependent events?

Answer 1

Independent = sampling WITH replacement (in the context of these types of card game analogies).

Dependent = sampling WITHOUT replacement.

So, in your case, you are sampling without replacement, because you and your friend " ... pull out one card, and then another, without putting the first one back."

When the events are dependent, you have to account for that fact in the second part of the probability equation.

Answer 2

The probability that you win is the probability that you pull two spades in a row. We can determine this probability directly by counting

P(win) = (# of choices for two spades in a row)/(# of choices for any two cards in a row)

The denominator is

# of choices for any two cards in a row = 52 * 51

The numerator is

# of choices of two spades in a row = (# of choices for first spade) * (# of choices for second spade)

= 13 * 12

So the final probability is

P(win) = (13 * 12) / (52 * 51) = 1/17 ≈ 0.059

In this case, the choices aren't independent because the possibilities for the second card are affected by what happens with the first card, specifically if the first card is a spade then there are 12 spades to choose from for the second card, but if the first card is not a spade then there are 13 spades to choose from for the second.