Question

1. a standard deck of cards contains 52 cards. there are four suits: diamonds, hearts, clubs, 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: Define events

Let $A$ be the event that the first card is a heart and $B$ be the event that the second card is a king.

Step2: Calculate $P(A)$

There are 13 hearts in a 52 - card deck. So $P(A)=\frac{13}{52}=\frac{1}{4}$.

Step3: Calculate $P(B)$

There are 4 kings in a 52 - card deck. So $P(B)=\frac{4}{52}=\frac{1}{13}$.

Step4: Calculate $P(A\cap B)$

The probability that the first card is a heart and the second card is a king. The probability of getting a heart first is $\frac{13}{52}$, and the probability of getting a king second (with replacement) is $\frac{4}{52}$. So $P(A\cap B)=\frac{13}{52}\times\frac{4}{52}=\frac{1}{52}$.

Step5: Use the addition - rule for probability

The addition rule for probability is $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
Substitute the values: $P(A\cup B)=\frac{1}{4}+\frac{1}{13}-\frac{1}{52}$.
First, find a common denominator, which is 52. Then $\frac{1}{4}=\frac{13}{52}$ and $\frac{1}{13}=\frac{4}{52}$.
So $P(A\cup B)=\frac{13 + 4-1}{52}=\frac{16}{52}=\frac{4}{13}$.

Answer:

$\frac{4}{13}$