What did his critics misunderstand about the assumptions Laplace was making to do this calculation

1 Suppose the parameter T�
 can take the values in the table below with the corresponding probabilities: Value Probability 10 0.45 20 0.05 30 0.18 40 0.01 1,000,000 0.31 a) Calculate the mean, mode, and median of the distribution of T�
. b) Which single value would you use to summarize the distribution if you had to? What would be the advantages and disadvantages of doing so?

2 .1 Suppose you and a friend play the following game: Your friend picks a number N from 1 to 6 and rolls N fair six-sided dice (that you’re not allowed to see). Your friend then tells you only how many of those dice came up with a number other than a one or a six, and you have to guess the number of dice, N. Suppose you are initially indifferent to the possible values of N, and your friend tells you “Three dice came up with something other than a one or six.” What is your guess for N? What is the mean of your posterior probability distribution for N?

2.2 If your friend were to pick up only those dice that had not come up with a one or six the first time and roll them again, what probability would you assign to the

event: “On the second roll, all of the dice will come up with either a one or six.”)

3. One of the original examples Pierre-Simon Laplace gave of his Rule of Succession was the “sunrise problem”: Letting p�
 be the chance of the sun rising each day, after observing it having risen 10,000 days in a row, using the rule of succession we would say the probability of it rising tomorrow is 10001�/10002��≈�0.9999�

Laplace was harshly criticized for this, because it seems ridiculous to say that the sun rising is anything less than certain. What did his critics misunderstand about the assumptions Laplace was making to do this calculation?

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