1. **State the problem:** We have a lottery machine that outputs digits 0 through 9 with equal probability if fair. We tested it 200 times and recorded the frequency of each digit. We want to find: (a) The theoretical probability of getting a 0 or 5 assuming fairness. (b) The experimental probability of getting a 0 or 5 from the data. (c) Choose the true statement about the relationship between number of trials and closeness of experimental to theoretical probability. 2. **Formula and rules:** - Theoretical probability for an event = \frac{Number\ of\ favorable\ outcomes}{Total\ possible\ outcomes} - Experimental probability = \frac{Number\ of\ times\ event\ occurred}{Total\ trials} - The Law of Large Numbers states that as the number of trials increases, the experimental probability tends to get closer to the theoretical probability. 3. **Calculate theoretical probability (a):** - There are 10 digits (0 to 9), each equally likely if fair. - Probability of 0 or 5 = Probability(0) + Probability(5) = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = 0.2 4. **Calculate experimental probability (b):** - Number of times 0 appeared = 20 - Number of times 5 appeared = 25 - Total trials = 200 - Experimental probability = \frac{20 + 25}{200} = \frac{45}{200} Show cancellation: $$\frac{45}{200} = \frac{\cancel{5}9}{\cancel{5}40} = \frac{9}{40} = 0.225$$ 5. **Answer part (c):** - The true statement is: "The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability." **Final answers:** (a) Theoretical probability = 0.200 (b) Experimental probability = 0.225 (c) The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.