answer:For three dice, the probability of rolling a number other than 6 on one die is frac{5}{6}. The outcomes can vary from rolling zero 6's to three 6's. Using the binomial formula, we calculate each probability: - Probability of rolling zero 6's: left(frac{5}{6}right)^3 = frac{125}{216} - Probability of rolling exactly one 6: binom{3}{1} left(frac{1}{6}right) left(frac{5}{6}right)^2 = 3 times frac{1}{6} times frac{25}{36} = frac{75}{216} - Probability of rolling exactly two 6's: binom{3}{2} left(frac{1}{6}right)^2 left(frac{5}{6}right) = 3 times frac{1}{36} times frac{5}{6} = frac{15}{216} - Probability of rolling exactly three 6's: left(frac{1}{6}right)^3 = frac{1}{216} Now, calculate the expected value, where each outcome is weighted by its probability: E = 0 cdot frac{125}{216} + 1 cdot frac{75}{216} + 2 cdot frac{15}{216} + 3 cdot frac{1}{216} = 0 + frac{75}{216} + frac{30}{216} + frac{3}{216} = frac{108}{216} = frac{1}{2}. Thus, the expected number of 6's obtained is boxed{frac{1}{2}}.

answer:Step 1: Extend Lines and Identify Angles Extend overline{PQ} and overline{RS} to meet at point T. Given angle PQS = 30^circ and angle QRS = 60^circ, calculate angle PTS and angle QTS: - angle PTS = 180^circ - angle QRS = 180^circ - 60^circ = 120^circ. - angle QTS = angle PQS = 30^circ. Step 2: Calculate angle STQ Using the angle sum property in triangle TSQ, we find: [ angle STQ = 180^circ - angle PTS - angle QTS = 180^circ - 120^circ - 30^circ = 30^circ. ] Since angle STQ = angle QTS = 30^circ, triangle TSQ is isosceles with TQ = QS = 1. Step 3: Use Similarity of Triangles Since overline{PQ} parallel overline{RS}, triangle TSQ sim triangle TQP. The ratio of similarity between triangle TSR and triangle TQP is given by the ratio of their parallel sides: [ frac{RS}{PQ} = frac{7}{4}. ] Since SQ = QS = 1, express RS in terms of PR and SQ: [ RS = PR + SQ = PR + 1. ] Step 4: Solve for PR Using the similarity ratio: [ frac{7}{4} = frac{RS}{PQ} = frac{PR + 1}{1} = PR + 1. ] Solving for PR: [ PR = frac{7}{4} - 1 = frac{7}{4} - frac{4}{4} = frac{3}{4}. ] Conclusion Thus, the length of PR is frac{3{4}}. The final answer is C) boxed{frac 34}

answer:Let's denote the number of times John goes to the gym each week as ( x ). Each day he goes to the gym, he spends 1 hour lifting weights. Additionally, he spends a third of that time warming up and doing cardio. So, the time he spends on cardio and warming up each day is ( frac{1}{3} times 1 ) hour, which is ( frac{1}{3} ) hour. Therefore, the total time he spends at the gym each day is the sum of the weightlifting time and the cardio/warm-up time: ( 1 ) hour (weightlifting) + ( frac{1}{3} ) hour (cardio/warm-up) = ( 1 + frac{1}{3} ) hours. To find the total time in hours, we need to convert ( frac{1}{3} ) to a decimal, which is approximately ( 0.333 ). So, the total time spent at the gym each day is ( 1 + 0.333 = 1.333 ) hours. Since he spends 4 hours at the gym per week, we can find the number of times he goes to the gym by dividing the total weekly gym time by the time spent per day: ( x = frac{4 text{ hours per week}}{1.333 text{ hours per day}} ). Now, let's calculate ( x ): ( x = frac{4}{1.333} approx 3 ). So, John goes to the gym approximately boxed{3} times a week.

answer:Jason had boxed{13} Pokemon cards to start with.