Independent Events of Playing Cards
Problem 732
A card is chosen randomly from a deck of the standard 52 playing cards.
Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart.
Prove or disprove that the events $E$ and $F$ are independent.
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Definition of Independence
Events $E$ and $F$ are said to be independent if
\[P(E \cap F) = P(E) \cdot P(F).\]
Intuitively, this means that an occurrence of one does not change the probability that the other occurs. Mathematically, this can be seen as follows.
If $P(E) \neq 0$, then independency implies
\[P(F \mid E) = \frac{P(F \cap E)}{P(E)} = \frac{P(F)P(E)}{P(E)} = P(F).\]
Solution.
We prove that the events $E$ and $F$ are independent.
First, we have $P(E \cap F) = 1/52$ because there is only one King of hearts card in the deck of 52 cards.
Since there are four kings, we have $P(E) = 4/52$. As there are $13$ heart cards, we have $P(F) = 13/52$.
Thus, we see that
\[P(E)P(F) = \frac{4}{52} \cdot \frac{13}{52} = \frac{1}{52} = P(E \cap F).\]
This implies that the events $E$ and $F$ are independent.
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