Question

a card is chosen at random from a standard deck (52 cards). find each probability as a fraction in simplest form. p(black or queen)

Explanation:

Step1: Recall the formula for probability of union

The formula for \( P(A \cup B) \) is \( P(A) + P(B) - P(A \cap B) \), where \( A \) is the event of choosing a black card and \( B \) is the event of choosing a queen.

Step2: Find \( P(A) \) (probability of black card)

In a standard deck, there are 26 black cards (13 spades and 13 clubs) out of 52. So \( P(A)=\frac{26}{52} \).

Step3: Find \( P(B) \) (probability of queen)

There are 4 queens (one for each suit) out of 52. So \( P(B)=\frac{4}{52} \).

Step4: Find \( P(A \cap B) \) (probability of black queen)

There are 2 black queens (queen of spades and queen of clubs) out of 52. So \( P(A \cap B)=\frac{2}{52} \).

Step5: Calculate \( P(A \cup B) \)

Substitute the values into the formula: \( P(A \cup B)=\frac{26}{52}+\frac{4}{52}-\frac{2}{52}=\frac{26 + 4- 2}{52}=\frac{28}{52} \). Simplify the fraction by dividing numerator and denominator by 4: \( \frac{28\div4}{52\div4}=\frac{7}{13} \).

Answer:

\(\frac{7}{13}\)