Question

Suppose that you have three fair 6-sided dice that you roll and then count the total. Estimate the mean total if you were to roll the dice over and over by simulating 250,000 rolls and computing the average of the random outcomes. (In Python)

Answer #1

Python code with comments

The code in text format

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import numpy as np

#Define the number of rolls

R=250000

#simulate R rolls of 3 dice each, making this a matrix of 3xR

x=np.random.randint(low=1,high=6+1,size=(3,R))

#get the R column sums, which is the total of 3 dice

s=x.sum(0)

#compute the average of these R totals

m=np.mean(s)

print('The estimated mean total of rolling 3 dice is %.4f'%m)

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You roll two fair six-sided dice. What is the probability that
the sum of the two dice values is exactly five? Be sure to count
all possible outcomes. (Hint: The event space has 36 distinct
outcomes).

Suppose you roll two twenty-sided dice, like the one I gave you
in class today. Let X1,X2 the outcomes of the rolls of these two
fair dice which can be viewed as a random sample of size 2 from a
uniform distribution on integers.
a) What is population from which these random samples are drawn?
Find the mean (µ) and variance of this population (σ2)? Use a Word
File to show your calculations and results.

Consider an experiment where we roll 7 fair 6-sided dice
simultaneously (the results of the dice are
independent from each other).
(a) What is the probability that exactly 3 of the dice are greater
than or equal to 5?
(b) Suppose now that each of the 7 6-sided dice are weighted the
same such that the probability of
rolling a 6 is 0.5, and every other side that is not a 6 has equal
probability of being rolled. If we...

1. Suppose you have a fair 6-sided die with the numbers 1
through 6 on the sides and a fair 5-sided die with the numbers 1
through 5 on the sides. What is the probability that a roll of the
six-sided die will produce a value larger than the roll of the
five-sided die?
2. What is the expected number of rolls until a fair five-sided
die rolls a 3? Justify your answer briefly.

We roll two fair 6-sided dice, A and B. Each one of the 36
possible outcomes is assumed to be equally likely.
a. Find the probability that dice A is larger than dice B.
b. Given that the roll resulted in a sum of 5 or less, ﬁnd the
conditional probability that the two dice were equal.
c. Given that the two dice land on different numbers, ﬁnd the
conditional probability that the two dice differed by 2.

Suppose that you roll 117 fair six-sided dice. Find the
probability that the sum of the dice is less than 400. (Round your
answers to four decimal places.)

Roll a 6-sided die and a 10-sided die. (Both dice are fair and
have an ace side.)
What is the chance that they will both land on
ace? (Give exact answer.)
What is the chance that neither will land on ace? (Give
exact answer.)
What is the chance that at least one will not land on
ace? (Give exact answer.)

Consider rolling a fair 6-sided dice.
Which of the following statements are correct?
Group of answer choices
The probability that it lands on a 1 is 1/6.
The probability of an even number on one roll of a dice is
2/6.
The probability of an even number on one roll of a dice is
3/6.
If we roll this dice a large number of times, then for about 5/6
of the time, it will NOT land on a 2.
Suppose...

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sum value appears that is 7 or less, and then we stop
afterwards.
a. What is the probability that exactly three
(pairs of) rolls are required?
b. What is the probability that at least three
(pairs of) rolls are needed?
c. What is the probability that, on the last
rolled pair, we get a result of exactly 7?

We roll three fair six-sided dice.
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(b) What is the probability that we roll a sum of at least
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(c) Now we roll three fair dice n times. How large need n be in
order to guarantee a better
than 50% chance of rolling a sum of at least 15, at least once?

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