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
\[P(E) = P(E \cap F) + P(E \cap F^c).\]
As $E$ and $F$ are independent, we know that
\[P(E \cap F) = P(E)\cdot P(F)\]
Combining these two equalities, we get
P(E \cap F^c) &= P(E) – P(E \cap F)\\
&= P(E) – P(E) \cdot P(F)\\
Since $P(F^c) = 1 – P(F)$, we obtain the equality
\[P(E \cap F^c) = P(E)\cdot P(F^c),\]
which implies that $E$ and $F^c$ are independent.
We just proved that when $E$ and $F$ are independent events, then $E$ and the complement $F^c$ are independent.
Now, we apply this statement to the independent events $E$ and $F^c$. Then we see that the complements $E^c$ and $F^c$ are independent.
In conclusion, if two events are independent, then their complements are also independent.
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
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 […]
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 […]
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 […]
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?
Let $R_1$ be the event that the first ball is red and $R_2$ be the event that the […]
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?
The sample space $S$ is […]
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?
Let $E$ […]