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
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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 […]

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 […]

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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 […]

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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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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 $E$ […]

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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 […]