Let $E$ and $F$ be events. Then the conditional probability that $E$ occurs given $F$ has occurred, which is denoted by $P(E\mid F)$, is defined as
\[P(E \mid F) = \frac{P(E \cap F)}{P(F)}.\]

For uniform space (such as fair die rolling), this can be rewritten as
\[P(E \mid F) = \frac{|E \cap F|}{|F|}.\]
Here the notation $|A|$ means the number of elements in $A$.

Solution (1)

Let $E = \{2, 3, 5\}$ be the set of prime number faces of a die. Let $F = \{1, 3, 5\}$ be the set of odd faces of a die. Then the required conditional probability can be calculated as follows:
\begin{align*}
P(E \mid F) &= \frac{|E \cap F|}{|F|}\\
&= \frac{|\{2, 3, 5\} \cap \{1, 3, 5\}|}{|\{1, 3, 5\}|}\\
&= \frac{|\{3, 5\}|}{\{1,3,5\}|}\\
&= \frac{2}{3}
\end{align*}

Solution (2)

The required conditional probability is
\begin{align*}
P(\{1\} \mid E) &= \frac{|\{1\} \cap E|}{|E|}\\
&= \frac{|\{1\} \cap \{2, 3, 5\}|}{|\{2, 3, 5\}|}\\
&= \frac{|\emptyset|}{|\{2, 3, 5\}|}\\
&= \frac{0}{3} = 0
\end{align*}

Probability Problems about Two Dice
Two fair and distinguishable six-sided dice are rolled.
(1) What is the probability that the sum of the upturned faces will equal $5$?
(2) What is the probability that the outcome of the second die is strictly greater than the first die?
Solution.
The sample space $S$ is […]

Overall Fraction of Defective Smartphones of Three Factories
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. Determine the overall fraction of […]

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 […]

Independent Events of Playing Cards
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.
Definition of Independence
Events […]

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 […]

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 […]

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 […]