Cards

Michal is solving a homework assignment:

What is the probability that when drawing two cards from a deck of $32$ cards, the drawn cards will include an ace or a king? While checking his solution, he found out that his classmates solved the problem as well, but each of them approached it differently. Who solved the problem correctly? (Among the $32$ cards, there are $8$ favorable cards (aces/kings) and $24$ unfavorable cards.)

Michal:

• First, we determine the probability that none of the drawn cards will be favorable.
• The probability of choosing the first card from the unfavorable cards is $\frac{24}{32}$.
• The probability of choosing the second card from the unfavorable cards is $\frac{23}{31}$.
• The probability that neither of the cards will be an ace or a king is $\frac{24}{32}\cdot\frac{23}{31}\cong0.5565$.
• The probability that the drawn cards will include an ace or a king is $1-\frac{24}{32}\cdot\frac{23}{31}\cong0.4435$.

Martina:

• We determine the probability that both drawn cards are favorable.
• The probability of choosing the first card from the favorable cards is $\frac{8}{32}$.
• The probability of choosing the second card from the favorable cards is $\frac{7}{31}$.
• The probability that the drawn cards include an ace or a king is $\frac{8}{32}\cdot\frac{7}{31}\cong0.0565$.

Pavel:

• We need to calculate the probability that when drawing two cards from the deck, both are favorable.
• The probability of choosing the first card from the favorable cards is $\frac{8}{32}$.
• The probability of choosing the second card from the favorable cards is $\frac{7}{31}$.
• The probability that the drawn cards will include an ace or a king is $\frac{8}{32}+\frac{7}{31}\cong0.4758$.

Martin:

• We determine the probability that both drawn cards are favorable.
• The probability of choosing the first card from the favorable cards is $\frac{8}{32}$.
• The probability of choosing the second card from the favorable cards is $\frac{7}{31}$.
• The probability that both cards will be aces or kings is $\frac{8}{32}\cdot\frac{7}{31}\cong0.0565$.
• The probability that the drawn cards will include an ace or a king is $1-\frac{8}{32}\cdot\frac{7}{31}\cong0.9435$.