Question

1. a standard deck of cards contains 52 cards. there are four suits: diamonds, hearts, clubs, and spades. each suit has 13 cards: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. you randomly choose two cards from a standard deck of cards with replacement. what is the probability that the first card is a heart or the second card is a king?

Explanation:

Step1: Calculate probability of first - card being a heart

The number of hearts in a deck is 13. The total number of cards is 52. So the probability that the first card is a heart, $P(H_1)=\frac{13}{52}=\frac{1}{4}$.

Step2: Calculate probability of second - card being a king

The number of kings in a deck is 4. The total number of cards is 52. So the probability that the second card is a king, $P(K_2)=\frac{4}{52}=\frac{1}{13}$.

Step3: Calculate probability of first - card being a heart and second - card being a king

Since the draws are with replacement, the events are independent. The probability that the first card is a heart and the second card is a king is $P(H_1\cap K_2)=P(H_1)\times P(K_2)=\frac{13}{52}\times\frac{4}{52}=\frac{1}{52}$.

Step4: Use the addition rule of probability

The addition rule for two events $A$ and $B$ is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Here, $A$ is the event that the first card is a heart and $B$ is the event that the second card is a king. So $P(H_1\cup K_2)=P(H_1)+P(K_2)-P(H_1\cap K_2)=\frac{1}{4}+\frac{1}{13}-\frac{1}{52}$.
First, find a common denominator (52): $\frac{1}{4}=\frac{13}{52}$ and $\frac{1}{13}=\frac{4}{52}$. Then $\frac{13}{52}+\frac{4}{52}-\frac{1}{52}=\frac{13 + 4-1}{52}=\frac{16}{52}=\frac{4}{13}$.

Answer:

$\frac{4}{13}$