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Independent events

Two events E and F are said to be independent if the occurrence of E makes it neither more nor less probable that F occurs and, conversely, if the occurrence of F makes it neither more nor less probable that E occurs. In other words, after receiving the information that E will happen, we revise our assessment of the probability that F will happen, computing the conditional probability of F given E; if F is independent of E, the probability of F remains the same as it was before receiving the information:[eq1]Conversely,[eq2]

In standard probability theory, rather than characterizing independence by the above two properties, independence is characterized in a more compact way, as follows.

Definition Two events E and F are independent if and only if[eq3]

It is easy to prove that this definition implies properties (1) and (2) above. In fact, suppose E and F are independent and (say) [eq4]. Then,[eq5]

Example An urn contains four balls $B_{1}$, $B_{2}$, $B_{3}$ and $B_{4}$. We draw one of them at random. The sample space is[eq6]Each of the four balls has the same probability of being drawn, equal to $frac{1}{4}$, i.e.,[eq7]Define the events E and F as follows:[eq8]Their respective probabilities are[eq9]The probability of the event $Ecap F~$is[eq10]Hence,[eq11]As a consequence, E and F are independent.

Mutually independent events

The definition of independence can be extended also to collections of more than two events.

Definition Let $E_{1}$, ..., $E_{n}$ be n events. $E_{1}$, ..., $E_{n}$ are jointly independent (or mutually independent) if and only if for any sub-collection of k events ($kleq n$) $E_{i_{1}}$, ..., $E_{i_{k}}$:[eq12]

Let $E_{1}$, ..., $E_{n}$ be a collection of n events. It is important to note that even if all the possible couples of events are independent (i.e., $E_{i}$ is independent of $E_{j}$ for any $j
eq i$), this does not imply that the events $E_{1}$, ..., $E_{n}$ are jointly independent. This is proved with a simple counter-example.

Example Consider the experiment presented in the previous example (extracting a ball from an urn that contains four balls). Define the events E, F and $G$ as follows:[eq13]It is immediate to show that[eq14]Thus, all the possible couple of events in the collection E, F, $G$ are independent. However, the three events are not jointly independent. In fact,[eq15]

On the contrary, it is obviously true that if $E_{1}$, ..., $E_{n}$ are jointly independent, then $E_{i}$ is independent of $E_{j}$ for any $j
eq i$.

Zero-probability events and independence

If E is a zero-probability event, then E is independent of any other event F.

Proof

Note that[eq16]As a consequence, by the monotonicity of probability,[eq17]But [eq18], so [eq19]. Since probabilities cannot be negative, it must be [eq20]. The latter fact implies independence:[eq21]

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Suppose that we toss a die. Six numbers (from 1 to $6)$ can appear face up, but we do not yet know which one of them will appear. The sample space is[eq22]Each of the six numbers is a sample point and is assigned probability $frac{1}{6}$. Define the events E and F as follows:[eq23]Prove that E and F are independent.

Solution

The probability of E is[eq24]The probability of F is[eq25]The probability of $Ecap F$ is[eq26]E and F are independent because[eq27]

Exercise 2

A firm undertakes two projects, A and $B$. The probabilities of having a successful outcome are $frac{3}{4}$ for project A and $frac{1}{2}$ for project $B$. The probability that both projects will have a successful outcome is $frac{7}{16}$. Are the two outcomes independent?

Solution

Denote by E the event "project A is successful", by F the event "project $B$ is successful" and by $G$ the event "both projects are successful". The event $G$ can be expressed as[eq28]If E and F are independent, it must be that[eq29]Therefore, the two outcomes are not independent.

Exercise 3

A firm undertakes two projects, A and $B$. The probabilities of having a successful outcome are $frac{2}{3}$ for project A and $frac{4}{5}$ for project $B$. What is the probability that neither of the two projects will have a successful outcome if their outcomes are independent?

Solution

Denote by E the event "project A is successful", by F the event "project $B$ is successful" and by $G$ the event "neither of the two projects is successful". The event $G$ can be expressed as:[eq30]where $E^{c}$ and $F^{c}$ are the complements of E and F. Using De Morgan's law ([eq31]) and the formula for the probability of a complement, we obtain[eq32]By using the formula for the probability of a union, we obtain[eq33]Finally, since E and F are independent, we have that[eq34]

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