In Problems 39–42, use the given table, which lists six possible assignments of probabilities for tossing a coin twice, to answer the following questions.
Which of the assignments of probabilities should be used if tails is twice as likely to occur as heads?

Verified step by step guidance

1. Step 1: Define the probabilities of heads (H) and tails (T) based on the condition that tails is twice as likely as heads. Let the probability of heads be \(p\), then the probability of tails is \$2p$.
2. Step 2: Since the coin is tossed twice, the sample space consists of four outcomes: HH, HT, TH, and TT. The probabilities of these outcomes can be expressed in terms of \(p\) and \$2p$ as follows:
3. \[P(HH) = p \times p = p^2\]
4. \[P(HT) = p \times 2p = 2p^2\]
5. \[P(TH) = 2p \times p = 2p^2\]
6. \[P(TT) = 2p \times 2p = 4p^2\]
7. Step 3: Use the fact that the sum of all probabilities must equal 1:
8. \[p^2 + 2p^2 + 2p^2 + 4p^2 = 1\]
9. Simplify the equation:
10. \[9p^2 = 1\]
11. Step 4: Solve for \(p^2\) and then find \(p\) (the probability of heads).
12. Step 5: Calculate the probabilities for each outcome using the value of \(p\) and compare these probabilities to the assignments in the table to identify which assignment matches the condition that tails is twice as likely as heads.

Key Concepts

Sample Space and Outcomes

The sample space is the set of all possible outcomes of an experiment. For tossing a coin twice, the sample space includes HH, HT, TH, and TT. Understanding the sample space is essential to assign probabilities correctly to each outcome.

Probability Assignments and Validity

Probability assignments must be non-negative and sum to 1 across all outcomes. Each assignment in the table represents a different way to distribute probabilities over the sample space, and only valid assignments that meet these criteria can represent real-world scenarios.

Conditional Probability and Likelihood Ratios

When tails is twice as likely as heads, the probabilities must reflect this ratio. This means the total probability of outcomes with tails should be twice that of outcomes with heads. Understanding how to apply likelihood ratios helps identify the correct probability assignment.