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. Add to solve later

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. Add to solve later

More from my site

You may also like...

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Probability Jewelry Company Quality Test Failure Probability

A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings,...

Close