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How does replacing a drawn card change a dependent card-drawing experiment into an independent one?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In probability theory, drawing cards without replacement creates dependent events because the outcome of the first draw alters the composition of the deck, thereby changing the probability of subsequent draws.1

Replacing the drawn card restores the deck to its original state before the next draw occurs. Because the deck composition remains identical for every trial, the outcome of the first draw does not affect the probability of the second. Mathematically, two events $A$ and $B$ are independent if the conditional probability $P (B [1] [2] | A)$ is equal to the marginal probability $P (B)$ . By ensuring the conditions of the experiment remain constant through replacement, you satisfy this definition, as the occurrence of the first event provides no information that changes the likelihood of the second.23

Would you like to explore how this concept of independence relates to the binomial distribution?