# Aces and Hearts

Back to Probability Theory

### Problem

• What are the probabilities of randomly picking the following from a standard deck of 52 cards?
• the Ace of Hearts
• an Ace
• a Heart
• an Ace or a Heart
• neither an Ace nor a Heart
• a Red or Black card
• a Black Heart

### Solution

• Ace of Hearts
• With 52 cards, each equally likely drawn, the probability of drawing the ace of hearts is 1/52.
• An Ace
• With four aces in the deck there are four chances of picking an ace from 52 possibilities.  So the probability is 4/52.
• This calculation in effect uses the Special Disjunction Rule (SDR):
• P(AvB) = P(A) + P(B), where
• v means or and & means and
• A and B are mutually exclusive, i.e. P(A&B) = 0
• Since drawing an ace means drawing the ace of hearts (AH), the ace of diamonds (AD), the ace of clubs (AC) or the ace of spades (AS), the Special Disjunction Rule yields:
• P(AH v AD v AC v AS) = P(AH) + P(AD) + P(AC) + P(AS) = 1/52 + 1/52 + 1/52 + 1/52 = 4/52
• A Heart
• Per the Special Disjunction Rule
• P(AH v 2H v 3H v…v KH) = P(AH) + P(2H) + P(3H) +…+ P(KH) =  1/52 + 1/52 + 1/52 +…+ 1/52 = 13/52.
• The probability of drawing a heart is thus the sum of the probabilities of drawing particular hearts.
• An Ace or Heart
• The Special Disjunction Rule can’t be used here since aces and hearts aren’t mutually exclusive.
• Instead we use the General Disjunction Rule (GDR):
• P(AvB) = P(A) + P(B) – P(A&B)
• Which yields:
• P(Ace v Heart) = P(Ace) + P(Heart) – P(Ace & Heart) = 4/52 + 13/52 – 1/52 = 16/52
• GDR subtracts P(Ace & Heart) so the ace of hearts isn’t double counted.  In the Venn Diagram adding the probabilities of an ace (4/52) and a heart (13/52), per SDR, counts the ace-and-heart common area twice, yielding 17/52.  GDR subtracts 1/52, correcting SDR’s double-counting.
• Neither an Ace nor a Heart
• The probability of neither an Ace nor Heart is P(~Ace &~Heart).
•  P(~Ace & ~Heart) = P~(Ace v Heart), since (~Ace & ~Heart) is logically equivalent to ~(Ace v Heart)
• The probability that something is false is 1 minus the probability it’s true, per the Negation Rule
• P(~C) = 1 – P(C).
• ~ means it’s false that
• P(Ace v Heart) = 16/52, per the calculation above.
• So,  P(~Ace & ~Heart) = P~(Ace v Heart) = 1 – P(Ace v Heart) = 1 – 16/52 = 36/52.
• A Red or Black card
• P(Red or Black) = 1.0, since every card is one or the other, a result confirmed by SDR:
• P(Red or Black) = P(Red) + P(Black) = ½ + ½ = 1
• It’s thus certain you draw a red or black card
•  A Black Heart
• P(Black & Heart) = 0,  there being no such card.
• It’s thus impossible that you draw a black heart.