Question

You simultaneously roll 5 dice, and if all 5 dice are the same value, you win. If your dice are not all the same, you get to re-roll all 5 dice again. You get three tries in total. What is the probability of winning? That is, what is the probability that in your three tries, at least one of your rolls consists of five-of-a-kind?

Answer #1

A bar has a dice game that works as follows: You simultaneously
roll 5 dice, and if all 5 dice are the same value, you win. If your
dice are not all the same, you get to re-roll all 5 dice again. You
get three tries in total. What is the probability of winning? That
is, what is the probability that in your three tries, at least one
of your rolls consists of five-of-a-kind?

Suppose that we roll a pair of (6 sided) dice until the first
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?

Please follow the comment.
2. Roll two fair dice repeatedly. If the sum is ≥ 10, then you
win.
(a) What is the probability that you start by winning 3 times in
a row?
(b)What is the probability that after rolling the pair of dice 5
times you win exactly 3 times?
(c) What is the probability that the first time you win is
before the tenth roll (of the pair), but after the fifth?

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...

Problem 4)A simplified version of the dice game 10,000 is played
using 5 dice. The player rolls the 5 six-sided dice, each 1 that is
rolled, the player achieves a score of 100.
A) How many possible ways are there to roll i l's over these 5
dice? (Hint: use combinations)
B) The probability of any one dice rolling to a value of 1 is
1/5 (since each dice has six sides). Use the binomial probability
distribution. Calculate the probability...

Players A and B take turns at rolling two dice, starting with A.
The first person to get a sum of at least 9 on a roll of the two
dice wins the game.
Find the probability that A will win the game if:
(a) the game is just about to begin
(b) 8, 4, 3, 7 and 7 have already been rolled
(c) a draw is to be declared in the event of no-one winning
within 6 rolls.

Players A and B take turns at rolling two dice, starting with A.
The first person to get a sum of at least 9 on a roll of the two
dice wins the game. Find the probability that A will win the game
if:
(a) the game is just about to begin
(b) 8, 4, 3, 7 and 7 have already been rolled
(c) a draw is to be declared in the event of no-one winning
within 6 rolls.

In Yahtzee, 5 standard dice are rolled up to three times on a
given turn. A player can choose to save anywhere from 0 - 5 of the
dice between rolls 1 and 2 and between rolls 2 and 3, meaning that
the player can set aside certain dice before rolling the remaining
dice if they choose to do so. Suppose that a given player does not
take advantage of saving die rolls and instead rolls all 5 dice
each...

You roll two dice. Let Y be a random variable that is the larger
of the two rolls (or, if the rolls are the same, your random
variable is that value).
Write Y as a function from the outcome space to the real
number
What is the distribution of Y?
What is P(Y≥5)?

If you roll four (six-sided) dice, what is the probability that
at least one dice will be different from the other three? Leave
answer as a fraction

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