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.

Independent and Dependent Events of Three Coins Tossing
Suppose that three fair coins are tossed. Let $H_1$ be the event that the first coin lands heads and let $H_2$ be the event that the second coin lands heads. Also, let $E$ be the event that exactly two coins lands heads in a row.
For each pair of these events, determine whether […]

Complement of Independent Events are Independent
Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$.
Prove that $E$ and $F^c$ are independent as well.
Solution.
Note that $E\cap F$ and $E \cap F^c$ are disjoint and $E = (E \cap F) \cup (E \cap F^c)$. It follows that
\[P(E) = P(E \cap F) + P(E […]

Probability of Having Lung Cancer For Smokers
Let $C$ be the event that a randomly chosen person has lung cancer. Let $S$ be the event of a person being a smoker.
Suppose that 10% of the population has lung cancer and 20% of the population are smokers. Also, suppose that we know that 70% of all people who have lung cancer […]

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, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the […]

What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?
Four fair coins are tossed.
(1) What is the probability that all coins land heads?
(2) What is the probability that all coins land heads if the first coin is heads?
(3) What is the probability that all coins land heads if at least one coin lands […]

If a Smartphone is Defective, Which Factory Made It?
A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively.
If a smartphone of this model is […]

Pick Two Balls from a Box, What is the Probability Both are Red?
There are three blue balls and two red balls in a box.
When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red?
Solution.
Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]