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.

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(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 card is chosen randomly from a deck of the standard 52 playing cards.
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Definition of Independence
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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 […]

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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 […]

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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 fair six-sided die is rolled.
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