Problem 73 - Applied Statistics and Probability for Engineers

Question

Use the axioms of probability to show the following: (a) For any event \(E, P\left(E^{\prime}\right)=1-P(E)\). (b) \(P(\varnothing)=0\) (c) If \(A\) is contained in \(B,\) then \(P(A) \leq P(B)\).

Step-by-Step Solution

Answer

(a) Proven using complement rule, (b) via probability axiom, (c) by subset principle.

Step 1: Understand the Complement Rule

According to the Complement Rule of probability, for any event \(E\), the probability of the complement of \(E\) (denoted as \(E'\)) is equal to one minus the probability of \(E\). This can be expressed as \(P(E') = 1 - P(E)\). This is derived from the axiom that the probability of the entire sample space is 1, \(P(S) = 1\), and since \(E \cup E' = S\) and \(E \cap E' = \varnothing\), it follows that \(P(E) + P(E') = 1\), leading to \(P(E') = 1 - P(E)\).

Step 2: Prove Probability of Empty Set

According to the probability axioms, the probability of the empty set \(\varnothing\) is 0. The empty set represents an impossible event, which means it has no chance of occurring. Formalizing this, for any event \(A\), \(A \cap \varnothing = \varnothing\) and \(A \cup \varnothing = A\). Therefore, no additional probability is added by the empty set, i.e., \(P(A \cup \varnothing) = P(A)\). Since there's no change, \(P(\varnothing) = 0\).

Step 3: Prove Probability of Subsets

To demonstrate that if \(A\) is contained in \(B\), then \(P(A) \leq P(B)\), we rely on the Axiom of Probability which states that if an event \(A\) is a subset of another event \(B\), then \(B\) contains all outcomes of \(A\). Thus, \(B\) has at least the same or more probability of occurring than \(A\), since \(A \subseteq B\) implies \(A = A \cap B\). Therefore, by the addition law, \(P(B) = P(A) + P(B - A)\) (where \(B - A\) represents elements of \(B\) not in \(A\)) which implies \(P(A) \leq P(B)\).

Key Concepts

Complement Rule

The complement rule is a key concept in the field of probability. It revolves around the likelihood of an event not occurring. An event and its complement are all the possibilities within a sample space that do not belong to that event. As expressed mathematically, for an event \(E\), the probability of the complement \(E'\) is given by:\[ P(E') = 1 - P(E) \]This means, if you already know how likely event \(E\) is to happen, you can easily find out the chance of it not happening. This is because together, the event and its complement cover the entire sample space, which always has a total probability of 1. - **Sample Space**: All possible outcomes combined - **Event E**: The occurrence we are focusing on- **Complement E'**: All outcomes not in event E

The complement rule is extremely useful when probabilities are complex, but figuring out the opposite is simpler.

Empty Set Probability

In probability, the concept of the empty set, represented as \(\varnothing\), is fundamental. Although it might sound unfamiliar, it is just another way to describe an impossible event—something that has no chance of happening.The empty set has a probability of zero:\[ P(\varnothing) = 0 \]This arises from one of the axioms of probability that explains that no elements exist in the empty set to contribute any probability mass. Therefore, when no outcomes are possible, the probability is naturally zero. Here's how these properties are expressed:- **Intersection with Any Event**: \(A \cap \varnothing = \varnothing\)- **Union with Any Event**: \(A \cup \varnothing = A\)These properties highlight that the empty set doesn't change the probability of any real event \(A\) when it is involved in combinations.

Subset Probability

Understanding the probability relationship between a subset and its superset is also crucial. If you have two events, \(A\) and \(B\), and knowing that \(A\) is a subset of \(B\) implies the outcomes within \(A\) are also found in \(B\).This can be mathematically expressed as:\[ P(A) \leq P(B) \]The reasoning here is simple: since \(B\) includes every outcome of \(A\), plus possibly more, the probability of \(B\) occurring is at least as great as that of \(A\). Through the probability axioms and addition rules, this becomes clear:- **Subset Inclusion**: \(A \subseteq B\) implies all outcomes of \(A\) are in \(B\)- **Addition Law**: \(P(B) = P(A) + P(B - A)\)Using this concept helps us compare the likelihood of events, especially when one event is an expanded version of another.