Question

Let’s assume that there are two dice, and we will roll one of them, but we don’t know which one. The probability of rolling either dice is 1/2. One of them is fair in the sense that all 6 outcomes are equally likely. The other die gives probability 1/3 to numbers 1 through 3 and zero probability to numbers 4-6.

a-)The first roll was a 4. What is the probability that it was the fair die?

b-)The first roll was a 3. What is the probability that it was the fair die?

Hint(Bayes Rule)

Answer #1

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.

Consider rolling two fair six-sided dice.
a) Given that the roll resulted in sum of 8, find the
conditional probability that first die roll is 6.
b) Given that the roll resulted in sum of 4 or less, find the
conditional probability that doubles are rolled.
c) Given that the two dice land on different numbers, find the
conditional probability that at least one die is a 6.

Let’s say that you have three dice that are different colors;
one is blue, one is pink, and one is grey. Figure out the
probability of each of the following outcomes, showing as much work
as you can:
b) rolling a 3 or 4 on the pink die AND a 3 or 4 on the grey die
on the same roll

Rolling Snake Eyes
We roll two six-sided dice d1 and d2. What is
the probability that we roll two ones. (i.e., the probability that
d1=d2=1).
We roll two six-sided dice d1 and d2 and at
least one of the dice comes up as a one. What is the (conditional)
probability that we roll two ones.
We roll two six-sided dice d1 and d2 and at
least one of the dice comes up as a one. What is the (conditional)
probability...

We roll three fair six-sided dice.
(a) What is the probability that at least two of the dice land on a
number greater than 4?
(b) What is the probability that we roll a sum of at least
15?
(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?

Question I: You roll 2 fair dice, one red and one
green.
a) What is the probability that the red die does land on
4? does not land on 4?
b) What is the probability that either die lands on 4?
that neither die lands on 4?
c) What is the probability that the sum of the dice is
2? 3? 4? 5? 6? 7? 8? 9? 10? 11? 12?

QUESTION 5: A 4-sided die is used for some games. This die can
only land on the numbers 1, 2, 3, or 4. All these outcomes are
equally likely. Suppose we roll two 4-sided dice.
a. (3 pts) Write out the sample space (the set of all the ways
in which the two dice could land).
b. (2 pts) What is the probability that at least one of the dice
shows a 3?
c. (3 pts) Suppose we compute the...

Suppose you roll two six-sided dice at once, one yellow and the
other green.
Find the probability that: (1) The yellow die shows 3 and the
green shows
5. (2) The yellow die is even and the green shows 1. (3) The sum
of the
two numbers shown is 6.

We keep rolling 3 fair dice, a red die, a blue die, and a green
die, and write down the outcomes. We stop when all 6^3 = 216
possible outcomes show up at least once. What is the average
waiting time(=number of rolls)?

1. Game of rolling dice
a. Roll a fair die once. What is the sample space? What is the
probability to get “six”? What is the probability to get “six” or
“five”?
b. Roll a fair die 10 times. What is the probability to get
“six” twice? What is the probability to get six at
least twice?
c. Roll a fair die 10 times. What is the expected value and
variance of getting “six”?
d. If you roll the die...

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