Problem 48 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(E\) be any event in a sample space \(S .\) a. Are \(E\) and \(S\) independent? Explain your answer. b. Are \(E\) and \(\varnothing\) independent? Explain your answer.

Step-by-Step Solution

Answer

a. Yes, E and S are independent because $P(E \cap S) = P(E)P(S)$ holds as $P(E) = P(E)(1)$. b. Yes, E and $\varnothing$ are independent because $P(E \cap \varnothing) = P(E)P(\varnothing)$ holds as $0 = P(E)(0)$.

Step 1: a. Determine if E and S are independent

To determine if the event \(E\) and the sample space \(S\) are independent, we will use the definition of independence. We know that: - \(P(S) = 1\) since \(S\) represents the entire sample space. - \(P(E \cap S) = P(E)\) because \(E\) must be a subset of \(S\). Now, let's check if the equality holds for independence: \[P(E \cap S) = P(E)P(S)\] Substitute the values we know: \[P(E) = P(E)(1)\] \[P(E) = P(E)\] Since the equality holds, \(E\) and \(S\) are indeed independent events.

Step 2: b. Determine if E and ∅ are independent

To determine if the event \(E\) and the empty set \(\varnothing\) are independent, we will again use the definition of independence. We know that: - \(P(\varnothing) = 0\) since there is no outcome in the empty set. - \(P(E)\) can be any probability within the interval \((0, 1]\). - \(P(E \cap \varnothing) = P(\varnothing)\) since the intersection of any set with the empty set is the empty set. Now, let's check if the equality holds for independence: \[P(E \cap \varnothing) = P(E)P(\varnothing)\] Substitute the values we know: \[P(\varnothing) = P(E)(0)\] \[0 = 0\] Since the equality holds, \(E\) and \(\varnothing\) are indeed independent events.

Key Concepts

Independent Events

Understanding the concept of independent events is key to solving many probability problems. In probability theory, two events are considered independent if the occurrence of one does not affect the likelihood of the other occurring. This means that the probability of both events happening together is simply the product of their individual probabilities.

Mathematically, two events, say \(A\) and \(B\), are independent if:

• \(P(A \cap B) = P(A)P(B)\)

In our exercise scenario, we've established that if an event \(E\) is compared with the entire sample space \(S\), the two are independent. That's because the sample space includes all possible outcomes, including those in \(E\), so the occurrence of event \(E\) doesn't influence or change the composition of \(S\). This inherent inclusion means the equation for independence holds: \(P(E \cap S) = P(E)P(S)\).
On the other hand, comparing an event \(E\) with an empty set \(\varnothing\), we find they are also independent. Since the empty set contains no outcomes (i.e., \(P(\varnothing) = 0\)), the intersection of \(E\) with \(\varnothing\) is always an empty set. Thus, the probability conditions align: \(P(E \cap \varnothing) = 0 = P(E) \times 0\).

Recognizing when events are independent helps in accurately calculating their combined probabilities, crucial for problem-solving in statistics and probability.

Sample Space

The concept of a sample space is foundational in probability. A sample space, often denoted by \(S\), represents the set of all possible outcomes in a probabilistic experiment. For example, when flipping a coin, the sample space is \{Heads, Tails\}.

In the context of our exercise, the sample space \(S\) inherently includes all imaginable outcomes, making its probability \(P(S)\) always equal to 1. This is because the occurrence of one or more events from the sample space is certain—there aren't any other outcomes outside \(S\).

Let's imagine a dice roll: the sample space \(S\) is \{1, 2, 3, 4, 5, 6\}. Here, the event \(E\) could be rolling an even number, \{2, 4, 6\}. Since even outcomes are already part of the sample space, there’s a 100% chance that rolling the die results in one of the sample space's elements.

• In essence, \(P(S) = 1\), signifying certainty.

The sample space is integral for determining probabilities of events, as every other probability calculation depends on understanding the full spectrum of possible outcomes.

Empty Set

In probability, the empty set, represented as \(\varnothing\), plays a unique role. It is defined as a set with no elements, and its probability value is always zero: \(P(\varnothing) = 0\).

The empty set is significant because its presence signifies impossibility within a probability experiment. For example, if asked to pick a card from an empty hat, the event of picking a card is captured by \(\varnothing\), since there are no cards to draw from.

A key aspect to remember about the empty set is that any intersection involving \(\varnothing\), with any other set \(E\), results in \(\varnothing\). This means if you intersect \(E\) with \(\varnothing\), you get \(P(E \cap \varnothing) = P(\varnothing) = 0\). Thus, it holds that the empty set remains independent of any event based on the probability of intersections since:

• \(P(E \cap \varnothing) = 0 = P(E) \times P(\varnothing)\)

Recognizing the function of the empty set in a sample space aids in identifying impossible events and simplifying complex probability problems.