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Chapter 6: Problem 9

List the elements in each of the sets. The set of all outcomes of tossing a pair of (a) distinguishable coins (b) indistinguishable coins.

Short Answer

Expert verified
The set of all outcomes for distinguishable coins is {(H1, H2), (H1, T2), (T1, H2), (T1, T2)}. The set of all outcomes for indistinguishable coins is {2H, 1H & 1T, 2T}.

Step by step solution

01

Identify the possible outcomes for distinguishable coins

When tossing a pair of distinguishable coins, there are two coins with two different sides for each. We'll denote the coins and their outcomes as follows: Coin 1 can be either heads (H1) or tails (T1), and Coin 2 can be either heads (H2) or tails (T2).
02

List the elements for distinguishable coins

Now we will list all possible combinations of the outcomes for the two distinguishable coins: - Coin 1 is heads (H1) and Coin 2 is heads (H2): (H1, H2) - Coin 1 is heads (H1) and Coin 2 is tails (T2): (H1, T2) - Coin 1 is tails (T1) and Coin 2 is heads (H2): (T1, H2) - Coin 1 is tails (T1) and Coin 2 is tails (T2): (T1, T2) Therefore, the set of all outcomes for distinguishable coins is {(H1, H2), (H1, T2), (T1, H2), (T1, T2)}.
03

Identify the possible outcomes for indistinguishable coins

When tossing a pair of indistinguishable coins, we cannot differentiate between the two coins, so we need to consider the outcomes based on how many heads (H) and tails (T) appear in total.
04

List the elements for indistinguishable coins

Here are all the possible combinations for indistinguishable coins: - Both coins are heads: 2H - One coin is heads and the other coin is tails: 1H, 1T - Both coins are tails: 2T Therefore, the set of all outcomes for indistinguishable coins is {2H, 1H & 1T, 2T}.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distinguishable Coins
When working with distinguishable coins, we consider each coin as unique. This means we can tell them apart, like a silver coin and a copper coin. Each coin has two sides: heads (H) and tails (T).

Let's say we have two coins, Coin 1 and Coin 2. If we toss them, we have several possible outcomes:
  • Both show heads: (H1, H2)
  • First is heads, second is tails: (H1, T2)
  • First is tails, second is heads: (T1, H2)
  • Both show tails: (T1, T2)
This illustrates that with distinguishable coins, we have 4 distinct outcomes. The order of the coins matters, making each combination unique.
Indistinguishable Coins
Indistinguishable coins mean we can't tell the coins apart. They look the same, like two identical quarters. Here, we focus on the result, not which coin shows what.

For two indistinguishable coins, the possible outcomes are:
  • Both heads: 2H
  • One head, one tail: 1H & 1T
  • Both tails: 2T
In this scenario, the order doesn't matter. So, an outcome of 1H & 1T is the same whether it鈥檚 heads on the first or second coin. This reduces the total number of unique results because some outcomes are equivalent.
Set Theory
Set theory provides a way to organize and explore different outcomes mathematically. It helps us list and compare different sets of outcomes like those from coin tosses.

When we list outcomes of distinguishable coins, we create a set with ordered pairs, for example, \( \{(H1, H2), (H1, T2), (T1, H2), (T1, T2)\} \). This is called an ordered set because the sequence matters.
  • Ordered Pairs: Shows specific order, like Coin 1 heads and Coin 2 tails.
On the other hand, for indistinguishable coins, we use an unordered set. Here, the sequence is irrelevant. The set looks like \( \{2H, 1H \& 1T, 2T\} \).
  • Unordered Sets: Focus on quantity, not order.
Set theory is crucial in organizing such outcomes and visualizing their possibilities, helping us grasp the different scenarios clearly.

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Most popular questions from this chapter

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