Question

A fair penny and a fair nickel are tossed. Let \(A\) be the event "heads on the penny." Let \(B\) be the event "tails on the nickel." Let \(C\) be the event "the coins land the same way." (a) Choose a sample space \(\Omega,\) and represent \(A, B,\) and \(C\) in terms of \(\Omega\). (b) Which pairs of events chosen from \(A, B,\) and \(C\) are independent pairs? (c) Is the set of events \(\\{A, B, C\\}\) an independent set?

Short answer

(a) Sample space: \(\Omega = \{(H,H), (H,T), (T,H), (T,T)\}\). (b) No pairs are independent. (c) \(\{A, B, C\}\) is not independent.

Step-by-step solution

Define the Sample Space

Begin by listing all the possible outcomes when both a penny and a nickel are tossed. Each outcome can be represented as an ordered pair, where the first entry corresponds to the outcome of the penny and the second entry to the outcome of the nickel. The possible outcomes are: \[ \Omega = \{ (H, H), (H, T), (T, H), (T, T) \} \] where \(H\) represents heads and \(T\) represents tails.

Define Events A, B, and C

Now, interpret the events based on the defined sample space:- Event \(A\): "heads on the penny", which includes outcomes where the penny shows heads. Hence, \[ A = \{ (H, H), (H, T) \} \].- Event \(B\): "tails on the nickel", including outcomes where the nickel shows tails. So, \[ B = \{ (H, T), (T, T) \} \].- Event \(C\): "the coins land the same way", includes outcomes where both coins show the same side. Thus, \[ C = \{ (H, H), (T, T) \} \].

Determine Pairwise Independence

To determine if events are independent, check if the probability of their intersection equals the product of their individual probabilities:- Calculate probabilities: \[ P(A) = \frac{2}{4} = \frac{1}{2}, \quad P(B) = \frac{2}{4} = \frac{1}{2}, \quad P(C) = \frac{2}{4} = \frac{1}{2} \]- Intersection and independence: - \(P(A \cap B) = \{(H, T)\} = \frac{1}{4}\); is \(\frac{1}{4} = \frac{1}{2} \times \frac{1}{2}?\) No. - \(P(A \cap C) = \{(H, H)\} = \frac{1}{4}\); is \(\frac{1}{4} = \frac{1}{2} \times \frac{1}{2}?\) No. - \(P(B \cap C) = \{(T, T)\} = \frac{1}{4}\); is \(\frac{1}{4} = \frac{1}{2} \times \frac{1}{2}?\) No.

Check for Independence of the Set of Events

A set of events \(\{A, B, C\}\) is independent if every possible intersection probability is equal to the product of individual probabilities:- From Step 3, each pair does not satisfy pair independence due to mismatching probabilities.- Therefore, the set \(\{A, B, C\}\) is not an independent set.

Key concepts

Sample Space

In probability theory, the sample space is the set of all possible outcomes of a random experiment. When dealing with a discrete scenario, such as tossing coins, we need to consider all the potential results.

For the exercise where a penny and a nickel are tossed, each coin can either land as heads (H) or tails (T). Therefore, the combination of outcomes from both coins forms the sample space. This is represented as an ordered pair where the first entry denotes the penny and the second the nickel.

Given this setup, the sample space, denoted by \( \Omega \), includes the following outcomes:

• \((H, H)\): Heads on both the penny and nickel.
• \((H, T)\): Heads on the penny, tails on the nickel.
• \((T, H)\): Tails on the penny, heads on the nickel.
• \((T, T)\): Tails on both the penny and nickel.

Understanding the sample space is crucial as it provides the foundation upon which we determine the probabilities of different events.

Independent Events

Independent events in probability are those whose occurrence does not influence each other. In simpler terms, the outcome of one event does not affect the probability of another. When examining independent events, their combined occurrence probability is the product of their individual probabilities.

Using the events defined in the exercise:

• Event \( A \): Heads on the penny
• Event \( B \): Tails on the nickel
• Event \( C \): Coins landing the same way

To check if any pair of these events is independent, verify if the probability of their intersection equals the product of their probabilities. However, as shown in the solution, none of the event pairs \((A, B), (A, C), \text{and} (B, C)\) satisfy this condition. Thus, these events are not independent. Understanding independent events helps in simplifying complex problems where multiple events occur together.

Intersection of Events

The intersection of events involves finding the outcomes common to multiple events. In probability, the intersection of two events, say \(A\) and \(B\), is denoted by \(A \cap B\) and refers to outcomes that satisfy both events occurring simultaneously.

In this exercise:

• For events \(A\) and \(B\), the intersection \(A \cap B\) includes outcomes where both heads show up on the penny and tails on the nickel, resulting in \(\{(H, T)\}\).
• Between \(A\) and \(C\), intersecting means both heads must be on the penny, with both coins showing heads, leading to \(\{(H, H)\}\).
• Lastly, for \(B\) and \(C\), tails show up on the nickel with both coins showing tails, forming \(\{(T, T)\}\).

The intersection of events is a valuable concept as it allows us to compute probabilities of simultaneous occurrences in a structured manner. It also informs whether these events are independent by comparing their intersections to the expected probability products if they were independent.