Question

Suppose that $E_1,E_2,\dots ,E_k$ are events in the same sample space and that some pair $E_i,E_j$ of these events are disjoint. (a) If all the events have positive probability, can the set $\text{Missing left or extra right}$ be an independent set of events? Explain your answer. (b) If one or more of the events has 0 probability, can the set $\text{Missing left or extra right}$ be an independent set of events?

Short answer

(a) No, if all events have positive probability, the set cannot be independent due to the contradiction with disjoint events. (b) Yes, if some events have zero probability, the set can be independent.

Step-by-step solution

Understanding Disjoint Events

Disjoint events are events that cannot occur simultaneously. If two events, such as $E_i$ and $E_j$, are disjoint, then their intersection $E_i\cap E_j$ is empty, meaning that $P(E_i\cap E_j)=0$.

Understanding Independent Events

Events are independent if the occurrence of one event does not affect the probability of the other. For events $E_i$ and $E_j$, they are independent if $P(E_i\cap E_j)=P(E_i)\times P(E_j)$.

Analyzing Part (a)

Given that the events have positive probability, if $E_i$ and $E_j$ are independent, then $P(E_i\cap E_j)=P(E_i)\times P(E_j)>0$. However, if they are also disjoint, $P(E_i\cap E_j)=0$, which contradicts independence as the product of positive probabilities is positive. Thus, the set $\{E_1,E_2,\dots ,E_k\}$ cannot be independent if all events have positive probability.

Considering Zero Probability in Part (b)

If any event has zero probability, for instance, $P(E_i)=0$, then $P(E_i\cap E_j)=P(E_i)\times P(E_j)=0$ will be satisfied even if $E_i$ and $E_j$ are disjoint. This is because any number times zero is zero, which does not contradict independence. Therefore, the set can be independent if one or more events have zero probability.

Key concepts

Disjoint Events

In probability theory, disjoint events are a key concept referring to events that cannot happen at the same time. Imagine you're choosing between different flavors of ice cream. If you eat chocolate, you can't simultaneously eat vanilla. That's disjoint! Mathematically, if you have two events, say $E_i$ and $E_j$, these are disjoint if their intersection is empty. This means $P(E_i\cap E_j)=0$.

Disjoint events are mutually exclusive, which means the occurrence of one excludes the occurrence of the other. This is crucial in probability calculations since for any two disjoint events, the probability of both occurring is always zero. Understanding this helps avoid common mistakes where one might erroneously add probabilities of non-disjoint events.

Independent Events

Independent events are another cornerstone in probability theory. They describe situations where the occurrence of one event does not influence the probability of the other. For example, flipping a coin and rolling a die are independent events; the result of one does not affect the other. Mathematically, events $E_i$ and $E_j$ are independent if $P(E_i\cap E_j)=P(E_i)\times P(E_j)$.

Independence is a stronger condition than you might think because it implies a specific relationship between probabilities. When calculating probabilities with independent events, multiplication delivers the joint probability. Misunderstanding this can lead to errors in figuring compound probabilities where one assumes dependencies that do not exist. So always check for independence in your computations!

Zero Probability

Events with zero probability are much like finding an empty jar when reaching for cookies – unexpected and somewhat disappointing. In probability, this means there's no chance of the event happening. When an event has zero probability, denoted as $P(E_i)=0$, it suggests that this outcome is impossible given the conditions of the experiment.

Zero probability plays an intriguing role in independence. If a set of events includes at least one event with zero probability, it can technically still be considered independent even if others are disjoint because multiplying by zero satisfies the condition $P(E_i\cap E_j)=P(E_i)\times P(E_j)=0$. This helps when certain events are impossible, yet you want to maintain an independence assumption for modeling other aspects.

Positive Probability

Positive probability involves events that have a greater than zero chance of occurring. Let's say each time you reach into that cookie jar, there's a real chance of finding a cookie. This is quite different from zero probability as it signifies actual likelihood.

In terms of independence, positive probabilities are essential when determining whether sets of events are actually independent. For events to be independent and not disjoint, their joint probability $P(E_i\cap E_j)$ must not equal zero if both $P(E_i)$ and $P(E_j)$ are positive. If events are disjoint, the intersection is zero, which contradicts positive independence since $P(E_i)\times P(E_j)$ will always be a positive number. Therefore, recognizing positive probability ensures proper handling of independence in calculations.