Question

2 set venn diagrams: deck of cards
a deck of cards is categorized by whether they are black cards, and whether they are 10
or greater cards. fill in the venn diagram to show how many cards are in each category.
then answer the additional questions below.
note: we will not consider aces as 10 or greater.
view image of cards: deck of cards
note: we will not consider aces as 10 or greater.
enter your results below as reduced fractions.
what fraction of all the cards are black?
what fraction of all the cards are 10 or greater?
what fraction of all the cards are black and 10 or greater?
what fraction of all the cards are black or 10 or greater?
what fraction of all the cards are neither black nor 10 or greater?

Explanation:

Step1: Calculate fraction of black cards

Total cards \( n = 52 \). Black cards (clubs and spades) \( n_{black}=26 \).
Fraction \(=\frac{26}{52}=\frac{1}{2}\).

Step2: Calculate fraction of 10 or greater cards

10, J, Q, K in each suit. 4 suits. So \( n_{10 +}=4\times4 = 16 \).
Fraction \(=\frac{16}{52}=\frac{4}{13}\).

Step3: Calculate fraction of black and 10 or greater cards

10, J, Q, K in clubs and spades. \( n_{black\cap10 +}=2\times4=8 \).
Fraction \(=\frac{8}{52}=\frac{2}{13}\).

Step4: Calculate fraction of black or 10 or greater cards

Use formula \( n(A\cup B)=n(A)+n(B)-n(A\cap B) \).
\( n_{black\cup10 +}=26 + 16-8=34 \).
Fraction \(=\frac{34}{52}=\frac{17}{26}\).

Step5: Calculate fraction of neither black nor 10 or greater cards

\( n_{\overline{black\cup10 +}}=52 - 34=18 \).
Fraction \(=\frac{18}{52}=\frac{9}{26}\).

Answer:

What fraction of all the cards are Black? \(\frac{1}{2}\)
What fraction of all the cards are 10 or greater? \(\frac{4}{13}\)
What fraction of all the cards are Black and 10 or greater? \(\frac{2}{13}\)
What fraction of all the cards are Black or 10 or greater? \(\frac{17}{26}\)
What fraction of all the cards are neither Black nor 10 or greater? \(\frac{9}{26}\)