Theoretical Probabilities 1. What is the theoretical probability of an unfavorable outcome? \( P(x \neq 1)= \) 1 \( ( \) fraction \( )= \) \( 1 / 3 \) \( \times \) \% (Round to the nearest percent.) 2. In the questions above, which is an exact answer? The fraction Which is the approximate answer? The percentage (NOTE: The percentages above are approximations because they were non-terminating and we had to round. Some percentages are exact answers. For example, \( \frac{1}{5}=0.20=20 \% \) ) 3. How do the fractions of the total outcomes and theoretical probability compare? The fractions of the total outcomes is the same as the theoretical probability \( \uparrow \) 4. What is the sum of the probabithies of the favorable outcomes and the unfavorable outcomes, \( P(x=1)+P(x \neq 1) \) ? \( \square \) 2 What is the percentage? \( \square \) 100 \%

Let's break down the problem step by step. ### 1. Theoretical Probability of an Unfavorable Outcome The theoretical probability of an unfavorable outcome is given as \( P(x \neq 1) = \frac{1}{3} \). To convert this fraction into a percentage, we can use the formula: \[ \text{Percentage} = \left( \frac{\text{Fraction}}{1} \right) \times 100 \] Substituting the fraction: \[ \text{Percentage} = \left( \frac{1}{3} \right) \times 100 \] Calculating this gives: \[ \text{Percentage} = \frac{100}{3} \approx 33.33\% \] Rounding to the nearest percent, we get: \[ \text{Rounded Percentage} = 33\% \] ### 2. Exact and Approximate Answers - **Exact Answer**: The fraction \( \frac{1}{3} \) is the exact answer. - **Approximate Answer**: The percentage \( 33\% \) is the approximate answer because it was rounded. ### 3. Comparison of Fractions of Total Outcomes and Theoretical Probability The fractions of the total outcomes and the theoretical probability are indeed the same. This is because the theoretical probability is defined as the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, we can conclude: \[ \text{The fractions of the total outcomes is the same as the theoretical probability} \uparrow \] ### 4. Sum of Probabilities of Favorable and Unfavorable Outcomes The sum of the probabilities of the favorable outcomes and the unfavorable outcomes is given by: \[ P(x=1) + P(x \neq 1) \] From the information provided, we know: - \( P(x \neq 1) = \frac{1}{3} \) - Since the total probability must equal 1, we can find \( P(x=1) \): \[ P(x=1) = 1 - P(x \neq 1) = 1 - \frac{1}{3} = \frac{2}{3} \] Now, summing these probabilities: \[ P(x=1) + P(x \neq 1) = \frac{2}{3} + \frac{1}{3} = 1 \] ### Percentage of the Sum of Probabilities The percentage of the sum of probabilities is: \[ \text{Percentage} = 1 \times 100 = 100\% \] ### Summary of Answers 1. \( P(x \neq 1) = \frac{1}{3} \) and rounded percentage is \( 33\% \). 2. Exact answer: The fraction \( \frac{1}{3} \); Approximate answer: The percentage \( 33\% \). 3. The fractions of the total outcomes is the same as the theoretical probability \( \uparrow \). 4. The sum of the probabilities \( P(x=1) + P(x \neq 1) = 1 \) and the percentage is \( 100\% \).