Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the second ball is red. Thus, the probability that both of the balls are red is given by $P(R_1 \cap R_2)$. By the product rule, we have
\[P(R_1 \cap R_2) = P(R_1) \cdot P(R_2 \mid R_1).\]

We have $P(R_1) = 2/5$ because we have a total of five balls of which two are red. Now, if the first ball is red, then the rest are three blue balls and one red ball. Hence the conditional probability $P(R_2 \mid R_1)$ is given by $P(R_2 \mid R_1) = 1/4$.

Combining these, we obtain
\[P(R_1 \cap R_2) = \frac{2}{5} \cdot \frac{1}{4} = \frac{1}{10}.\]
In conclusion, the probability that both of the balls are red is $1/10$.

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 […]

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 […]

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 […]

Conditional Probability Problems about Die Rolling
A fair six-sided die is rolled.
(1) What is the conditional probability that the die lands on a prime number given the die lands on an odd number?
(2) What is the conditional probability that the die lands on 1 given the die lands on a prime number?
Solution.
Let $E$ […]

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 […]

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 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 […]