Questions: What fraction of all the cards are Red or Face cards? What fraction of all the cards are neither Red nor Face cards?

Solution Steps

To solve these questions, we need to understand the composition of a standard deck of 52 playing cards. A standard deck has 26 red cards (13 hearts and 13 diamonds) and 12 face cards (4 jacks, 4 queens, and 4 kings). Some cards are both red and face cards, so we need to account for this overlap.

1. Calculate the fraction of cards that are either red or face cards using the principle of inclusion-exclusion.
2. Calculate the fraction of cards that are neither red nor face cards by subtracting the fraction of red or face cards from 1.

In a standard deck of 52 playing cards, we have:

• Red cards: \( R = 26 \)
• Face cards: \( F = 12 \)
• Red face cards: \( RF = 6 \)

Using the principle of inclusion-exclusion, the total number of cards that are either red or face cards is given by: \[ R \cup F = R + F - RF = 26 + 12 - 6 = 32 \]

The fraction of cards that are red or face cards is: \[ \text{Fraction}_{R \cup F} = \frac{R \cup F}{\text{Total Cards}} = \frac{32}{52} \approx 0.6154 \]

The fraction of cards that are neither red nor face cards is: \[ \text{Fraction}_{\text{Neither}} = 1 - \text{Fraction}_{R \cup F} = 1 - 0.6154 \approx 0.3846 \]

Final Answer