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Stat 20 Computing Probabilities Consider picking numbers from the following box. 0 1 2 3 Let 𝐴 be the event that the first pick yields an even number; 𝐵 be the event that the second pick is greater than or equal to one. 1. Pick two numbers without replacement. Find P (𝐵| first pick is 0 ) . 2. Pick two numbers without replacement. Find P (𝐵| first pick is 2 ) . 3. Pick two numbers with replacement. Find P (𝐵|𝐴) . Consider a fair, eight-sided die. 4. I roll the die four times. What is the probability that I roll the same number on all four rolls? 5. I roll the die twice. What is the probability that the rolls are different ? My dog Bella has two toys that she loves: an orange ball, and a thick rope. Each time she picks out a toy, she chooses it independently of all the other times (like a coin toss). That day, she was busy, so went to her toys three times. Define the events 𝐴 and 𝐵 where: 𝐴 is the event that she picked the rope at most one time; 𝐵 is the event that the toys she picked that day included both the rope and the ball. 6. Are 𝐴 and 𝐵 independent?

An American roulette wheel has 38 pockets, of which 18 are red, 18 black, and 2 are green. In each round, the wheel is spun and a white ball lands in one of these 38 pockets. 7. What is the probability of getting at the ball landing in a green pocket at least once in 5 spins of the wheel? A European roulette wheel has 37 pockets, of which 18 are red, 18 black, and only 1 green . The roulette wheel is numbered 0 through 36. 8. Write R code to simulate three spins of this wheel. 9. Now imagine that after each of the three spins, a pocket disappears. Simulate three spins of this magic wheel. We will now perform our first simulation of the year! For the following questions, consider the European roulette wheel of Question 7 and ensure your Quarto document will present the same results each time it is rendered. Write your code in the spaces below. 10. Create three vectors: one which contains 100 simulated spins of the European roulette wheel (call this one_hundred ), one which contains 1,000 such spins (call this one_thousand ), and another which contains 10,000 such spins (call this ten_thousand ). 11. Create a new vector that returns TRUE / FALSE values for each element in one_hundred , where TRUE means that the number spun is greater than 18, and save it. Repeat these steps for the one_thousand and ten_thousand vectors. 12. Find the proportion of numbers spun in each simulation that were greater than 18 (write the code and the proportion). Hint: how can you take a proportion of a logical vector? 13. Comment on how the proportions changed with respect to the true probability of spinning a number greater than 18 as the number of spins increased.

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