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.
Probabilities of An Infinite Sequence of Die Rolling
Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability.
(1) At least one die lands on the face 5 in the first $n$ rolls.
(2) Exactly $k$ dice land on the face 5 […]
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 […]
Probability that Alice Wins n Games Before Bob Wins m Games
Alice and Bob play some game against each other. The probability that Alice wins one game is $p$. Assume that each game is independent.
If Alice wins $n$ games before Bob wins $m$ games, then Alice becomes the champion of the game. What is the probability that Alice becomes the […]
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 […]