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 $1/10$.

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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 […]

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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 […]

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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 […]

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

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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 […]

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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 […]