Step 1

PROBABILITY RULES

A standard deck of cards contains 52 cards, of which 26 are red and 26 are black, 13 are of each suit (hearts, diamonds, spades, clubs) and of which 4 are of each denomination (A, 2 to 10, J, Q, K). The face cards are the jacks J, queens Q and kings K.

General multiplication rule:

\(\displaystyle{P}{\left({A}\cap{B}\right)}={P}{\left({A}\right)}\times{P}{\left({B}{\mid}{A}\right)}={P}{\left({B}\right)}\times{P}{\left({A}{\mid}{B}\right)}\)

Step 2

SOLUTION

a) 13 of the 52 caeds are hearts are hearts

The probability is the number of favorable outcomes divided by the number of possible outcomes:

\(\displaystyle{P}{\left({H}_{{{1}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{13}}}{{{52}}}}={\frac{{{1}}}{{{4}}}}\)

After one heart is selected, there are 12 hearts left among the remaining 51 cards.

\(\displaystyle{P}{\left({H}_{{{2}}}{\mid}{H}_{{{1}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{12}}}{{{51}}}}={\frac{{{4}}}{{{17}}}}\)

Use the general multiplication rule:

\(\displaystyle{P}{\left({H}_{{{1}}}\cap{H}_{{{2}}}\right)}={P}{\left({H}_{{{1}}}\right)}\times{P}{\left({H}_{{{2}}}{\mid}{H}_{{{1}}}\right)}={\frac{{{1}}}{{{4}}}}\times{\frac{{{4}}}{{{17}}}}={\frac{{{1}}}{{{17}}}}\approx{0.05882}\)

Step 3

b) 13 of the 52 cards are hearts

The probability is the number of favorable outcomes divided by the number of possible outcomes:

\(\displaystyle{P}{\left({H}_{{{1}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{13}}}{{{52}}}}={\frac{{{1}}}{{{4}}}}\)

After one heart is selected, there are 13 clubs left among the remaining 51 cards.

\(\displaystyle{P}{\left({C}_{{{2}}}{\mid}{H}_{{{1}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{13}}}{{{51}}}}\)

Use the general multiplication rule:

\(\displaystyle{P}{\left({H}_{{{1}}}\cap{C}_{{{2}}}\right)}={P}{\left({H}_{{{1}}}\right)}\times{P}{\left({C}_{{{2}}}{\mid}{H}_{{{1}}}\right)}={\frac{{{1}}}{{{4}}}}\times{\frac{{{13}}}{{{51}}}}={\frac{{{13}}}{{{204}}}}\approx{0.06373}\)

Step 4

c) 13 of the 52 cards are hearts, while 1 of these cards is the ace of hearts

The probability is the number of favorable outcomes divided by the number of possible outcomes:

\(\displaystyle{P}{\left({H}_{{{A}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{1}}}{{{52}}}}\)

\(\displaystyle{P}{\left({H}_{{\neg\ {A}}})={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{12}}}{{{52}}}}={\frac{{{3}}}{{{13}}}}\right.}\)

After the ace of hearts is selected, there are 3 aces left in the remaining 51 cards

\(\displaystyle{P}{\left({A}{\mid}{H}_{{{A}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{3}}}{{{51}}}}\)

After a heart that is not the ace of hearts is selected, there are 4 aces left in the remaining 51 cards

\(\displaystyle{P}{\left({A}{\mid}{H}_{{\neg\ {A}}}\right)}={\frac{{\#\text{of favorable outcomes}}}{{\#\text{of possible outcomes}}}}={\frac{{{4}}}{{{51}}}}\)

Use the general multiplication rule:

\(\displaystyle{P}{\left({H}\cap{A}\right)}={P}{\left({H}_{{{A}}}\cap{A}\right)}+{P}{\left({H}_{{\neg\ {A}}}\cap{A}\right)}\)

\(\displaystyle={P}{\left({H}_{{{A}}}\right)}\times{P}{\left({A}{\mid}{H}_{{{A}}}\right)}+{P}{\left({H}_{{\neg\ {A}}}\right)}\times{P}{\left({A}{\mid}{H}_{{\neg\ {A}}}\right)}\)

\(\displaystyle={\frac{{{1}}}{{{52}}}}\times{\frac{{{3}}}{{{51}}}}+{\frac{{{12}}}{{{52}}}}\times{\frac{{{4}}}{{{51}}}}\)

\(\displaystyle={\frac{{{1}}}{{{52}}}}\approx{0.01923}\)