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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 and E are independent events, the probability of F remains the same as it was before receiving the information:[eq1]Conversely,[eq2]

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Definition

In standard probability theory, rather than characterizing independence by properties (1) and (2) above, we define it in a more compact way, as follows.

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

It is easy to prove that this definition implies properties (1) and (2) above.

Proof

Suppose E and F are independent and (say) [eq4]. Then,[eq5]Note that we have assumed [eq6]. When [eq7], things are more complicated (see the discussion about division by zero in the lecture on conditional probability and in the references therein). It is exactly because of the difficulties that arise in defining [eq8] when [eq9] that a general definition of independence is not given by using properties (1) and (2).

Example

The following example shows how to check whether two events are independent in a simple probabilistic experiment.

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[eq10]Each of the four balls has the same probability of being drawn, equal to $frac{1}{4}$, i.e.,[eq11]Define the events E and F as follows:[eq12]Their respective probabilities are[eq13]The probability of the event $Ecap F~$is[eq14]Hence,[eq15]As a consequence, E and F are independent events.

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 ($kleq n$) $E_{i_{1}}$, ..., $E_{i_{k}}$:[eq16]

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:[eq17]It is immediate to show that[eq18]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,[eq19]

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[eq20]As a consequence, by the monotonicity of probability,[eq21]But [eq9], so [eq23]. Since probabilities cannot be negative, it must be [eq24]. The latter fact implies independence:[eq25]

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[eq26]Each of the six numbers is a sample point and is assigned probability $frac{1}{6}$. Define the events E and F as follows:[eq27]Prove that E and F are independent events.

Solution

The probability of E is[eq28]The probability of F is[eq29]The probability of $Ecap F$ is[eq30]E and F are independent events because[eq31]

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[eq32]If E and F are independent, it must be that[eq33]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:[eq34]where $E^{c}$ and $F^{c}$ are the complements of E and F. Using De Morgan's law ([eq35]) and the formula for the probability of a complement, we obtain[eq36]By using the formula for the probability of a union, we obtain[eq37]Finally, since E and F are independent, we have that[eq38]

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