Example 1.9-3 - The Four-Card Deck

A card deck contains four cards: AC, 2C, AD, and 2D. The deck is shuffled, and a pair of cards are dealt.^?And mistakes may be made.

With no further information, the unconditional sample space is:

		Second card
		AC	2C	AD	2D
First card	AC
	2C
	AD
	2D

There are 16 pairs, four of which are impossible;^?The deck does not contain duplicate cards, which rules out the same card as both cards in a pair. the sample-space size is 16 - 4 = 12.^?Or: there are four choices for the first card in a pair, and three choices for the second. 4*3 = 12.

If one of the cards in the pair is an ace, what's the probability that the other card is an ace too?

The condition that one of the cards in the pair is an ace eliminates pairs of twos from the sample space:

		Second card
		AC	2C	AD	2D
First card	AC
	2C
	AD
	2D

The sample-space size is now down to 10, and there are two pairs that contain only aces. The probability is 2/10 = 1/5 = 0.2.

This solution assumes too much. The question doesn't mention the order in which the cards are dealt,^?The question notes that one of the two cards is an ace, not that, for example, the first-dealt card is an ace. which means the pair (AC, 2D) is the same as the pair (2D, AC). This sample space is

{AC, AD} {2C, AD} {AD, 2D}

{AC, 2D} {2C, 2D}

{AC, 2C}

The condition rules out {2C, 2D}, and the question specifies {AC, AD} for a probability of 1/5.

Suppose one of the cards in the pair is AC. What is the probability that the other card is AD?

The stronger condition that one of the cards is AC eliminates more pairs from unconstrained sample space than the weaker condition that one of the cards is an ace:

		Second card
		AC	2C	AD	2D
First card	AC
	2C
	AD
	2D

The sample-space size is six, and there are two pairs that contain only aces. The probability is 2/6 = 1/3.

The unordered-pair conditional sample space is

{AC, AD} {2C, AD} {AD, 2D}

{AC, 2D} {2C, 2D}

{AC, 2C}

for a 1/3 probability that the other card is an ace.

This example shows that the stronger a condition is, the more of the sample space it may prune, which changes the probabilities assigned.