Complement of Independent Events are Independent

Probability problems

Problem 734

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.

LoadingAdd to solve later

Sponsored Links

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
\[P(E) = P(E \cap F) + P(E \cap F^c).\] As $E$ and $F$ are independent, we know that
\[P(E \cap F) = P(E)\cdot P(F)\] Combining these two equalities, we get
\begin{align*}
P(E \cap F^c) &= P(E) – P(E \cap F)\\
&= P(E) – P(E) \cdot P(F)\\
&= P(E)(1-P(F)).
\end{align*}

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.

Remark

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.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Probabilities of An Infinite Sequence of Die RollingProbabilities 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 […]
  • What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?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 […]
  • Independent Events of Playing CardsIndependent 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 […]
  • Independent and Dependent Events of Three Coins TossingIndependent 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 GamesProbability 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 ProbabilityJewelry 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 FactoriesOverall 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 […]
  • Lower and Upper Bounds of the Probability of the Intersection of Two EventsLower and Upper Bounds of the Probability of the Intersection of Two Events Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that \[a \leq P(A \cap B) \leq b\] and $P(A \cap B)$ could […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Probability
Probability problems
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...

Close