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.
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For each pair of these events, determine whether […]
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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.
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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.
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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?
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The pass […]