[20 points] Problem 1
To play a game, you have a box containing 20 fair four-sided dice, with faces {1,2,3,4}. This bag also contains 15 fair six-sided dice (faces {1,2,3,4,5,6}) and 5 fair twenty-sided dice (faces {1,2,3,4,...,19,20}). Call these 3 classes of die "Four", "Six" and "Twenty" (or F, S, and T, for short). You grab one die at random from the box.
Work the following problems by hand (and LaTeX your solutions) unless otherwise stated (but don't be afraid to simulate to check your result!).
Part A: You roll your die one time. What is the probability of the event 6R6, that you roll a 6? Include at least 4 decimal places in your final answer.
Solution:
Part B: Suppose you roll a 6. What is the probability that the die you chose from the box is a Six-sided die?
Solution:
Part C: Are the events 6R6 and S independent? Justify your answer using math.
Solution:
Part D: Suppose you roll your die n times and record each outcome. Obviously, if you roll 5 or 6 you know you did not pick a Four-sided die, and if you roll a 7 or higher, you must have picked a Twenty-sided die.
But, if you roll n times in a row and get only 1, 2, 3 or 4, you would have a stronger belief that you in fact grabbed one of the Four-sided dice. Let En be the event that you roll n rolls of 1, 2, 3 or 4 in a row. For example, one outcome in 3E3 is the sequence 2, 3, 1.
Given that you roll a single value of 1, 2, 3 or 4, what is the probability that you have grabbed a Four-sided die? That is, calculate by hand (1)P(FE1).
Solution:
Part E: First, work out by hand a formula for the probability in general that you are rolling a Four-sided die, given that you have rolled n values in a row that are 1, 2, 3 or 4 (that is, En). So you want a formula for ()P(FEn) in terms of n.
Then, define and code a function prob_four to calculate ()P(FEn). This function should only take one positive integer argument, n.
Verify that your function is performing as expected by checking your answer to Part D when =1n=1.
Solution:
In [ ]:
def prob_four(n): # Your code here.
Part F: What is the smallest value of n successive rolls of 1, 2, 3 or 4 that would make you believe that there's a better-than-90% chance that the die you grabbed is Four-sided? Plugging larger and larger values for n into your prob_four function is a very reasonable way to check this.
In [ ]:
# Your code here.
Solution:
[20 points] Problem 1 To play a game, you have a box containing 20 fair four-sided dice, with faces {1,2,3,4} . This bag also contains 15 fair six-sided dice (faces {1,2,3,4,5,6} ) and 5 fair twenty-sided dice (faces {1, 2, 3, 4,...,19,20} ). Call these 3 classes of die "Four", "Six" and "Twenty" (or F, S, and T, for short). You grab one die at random from the box. Work the following problems by hand and LaTeX your solutions) unless otherwise stated (but don't be afraid to simulate to check your result!). Part A: You roll your die one time. What is the probability of the event R6, that you roll a 6? Include at least 4 decimal places in your final answer. Solution Part B: Suppose you roll a 6. What is the probability that the die you chose from the box is a Six-sided die? Solution: Part C: Are the events Rand S independent? Justify your answer using math. Solution: Part D: Suppose you roll your die n times and record each outcome. Obviously, if you roll 5 or 6 you know you did not pick a Four-sided die, and if you roll a 7 or higher, you must have picked a Twenty-sided die. But, if you roll n times in a row and get only 1, 2, 3 or 4, you would have a stronger belief that you in fact grabbed one of the Four-sided dice. Let E, be the event that you roll n rolls of 1, 2, 3 or 4 in a row. For example, one outcome in Ez is the sequence 2, 3, 1. Given that you roll a single value of 1, 2, 3 or 4, what is the probability that you have grabbed a Four-sided die? That is, calculate by hand P( F E ). Solution: Part E. First, work out by hand a formula for the probability in general that you are rolling a Four-sided die, given that you have rolled n values in a row that are 1, 2, 3 or 4 (that is, En). So you want a formula for P( F E ) in terms of n. Then, define and code a function prob_four to calculate P(F ER). This function should only take one positive integer argument, n. Verify that your function is performing as expected by checking your answer to Part D when n=1. Solution def prob_four (n): # Your code here. Part F: What is the smallest value of n successive rolls of 1, 2, 3 or 4 that would make you believe that there's a better-than-90% chance that the die you grabbed is Four-sided? Plugging larger and larger values for n into your prob_four function is a very reasonable way to check this. # Pour code here. Solution: [20 points] Problem 1 To play a game, you have a box containing 20 fair four-sided dice, with faces {1,2,3,4} . This bag also contains 15 fair six-sided dice (faces {1,2,3,4,5,6} ) and 5 fair twenty-sided dice (faces {1, 2, 3, 4,...,19,20} ). Call these 3 classes of die "Four", "Six" and "Twenty" (or F, S, and T, for short). You grab one die at random from the box. Work the following problems by hand and LaTeX your solutions) unless otherwise stated (but don't be afraid to simulate to check your result!). Part A: You roll your die one time. What is the probability of the event R6, that you roll a 6? Include at least 4 decimal places in your final answer. Solution Part B: Suppose you roll a 6. What is the probability that the die you chose from the box is a Six-sided die? Solution: Part C: Are the events Rand S independent? Justify your answer using math. Solution: Part D: Suppose you roll your die n times and record each outcome. Obviously, if you roll 5 or 6 you know you did not pick a Four-sided die, and if you roll a 7 or higher, you must have picked a Twenty-sided die. But, if you roll n times in a row and get only 1, 2, 3 or 4, you would have a stronger belief that you in fact grabbed one of the Four-sided dice. Let E, be the event that you roll n rolls of 1, 2, 3 or 4 in a row. For example, one outcome in Ez is the sequence 2, 3, 1. Given that you roll a single value of 1, 2, 3 or 4, what is the probability that you have grabbed a Four-sided die? That is, calculate by hand P( F E ). Solution: Part E. First, work out by hand a formula for the probability in general that you are rolling a Four-sided die, given that you have rolled n values in a row that are 1, 2, 3 or 4 (that is, En). So you want a formula for P( F E ) in terms of n. Then, define and code a function prob_four to calculate P(F ER). This function should only take one positive integer argument, n. Verify that your function is performing as expected by checking your answer to Part D when n=1. Solution def prob_four (n): # Your code here. Part F: What is the smallest value of n successive rolls of 1, 2, 3 or 4 that would make you believe that there's a better-than-90% chance that the die you grabbed is Four-sided? Plugging larger and larger values for n into your prob_four function is a very reasonable way to check this. # Pour code here. Solution
