If we repeatedly roll a balanced die, then, in the long run, it will come up “4” about onesixth of the time. But what is the probability that such a die will come up “4” exactly once in five rolls? Please show work
Here,
n=5,p=1/6
The probability that die with come up '4' exactly once in five rolls

If we repeatedly roll a balanced die, then, in the long run, it will come up...
If we repeatedly toss a balanced coin, then, in the long run, it will come up heads about half the time. But what is the probability that such a coin will come up heads exactly half the time in 26 tosses?
We roll a fair die repeatedly. Let N be the number of rolls needed to see the first six, and let Y be the number of fives in the first N -1 rolls. In class, we saw that E[Y I N]- (N - 1)/5. Using this, find EiY]. Also, find Cov(Y, N). Hint: N -1 is a geometric random variable. (Why?)
Problem 5. A lopsided six-sided die is rolled repeatedly, with each roll being independent. The probabil- ity of rolling the value i is Pi, i = 1, … ,6. Let Xn denote the number of distinct values that appear in n rolls. (a) Find E|X, and E21 (b) What is the probability that in the n rolls of the dice, for n 2 3, a 1, 2, and 3 are each rolled at least once?
We roll a fair 8-sided die five times. (A fair 8-sided die is equally likely to be 1, 2, 3, 4, 5, 6, 7, or 8.) (a) What is the probability that at least one of the rolls is a 3? (b) Let X be the number of different values rolled. For example, if the five rolls are 2, 3, 8, 8, 7, then X = 4 (since four different values were rolled: 2,3,7,8). Find E[X].
Suppose you are rolling a fair four-sided die and a fair six-sided die and you are counting the number of ones that come up. What is the probability that both die roll ones? What is the probability that exactly one die rolls a one? What is the probability that neither die rolls a one? What is the expected number of ones? If you did this 1000 times, approximately how many times would you expect that exactly one die would roll...
If you roll a fair die 3 times, what is the probability that all 3 rolls will come up a value less than 4?
6. A six sided balanced die is rolled 30 times. The uppermost face is observed on each roll. A. Find the probability that each of the six sides shows up exactly five times. b. Find the probability that the die shows a 1 exactly five times and a 2 exactly 10 times.
problem 4 you repeatedly roll an ordinary eight-sided die five times. Let X equal the number of times you roll the die. Let Y equal the value of the first roll What is E[x] and E[Y]
We are going to play a game of chance. We will roll a die (half of a pair of dice) and if it comes up with one or two spots showing, you win and I will pay you $1.50. If it comes up with three, four, five, or six spots showing, you lose and you will pay me $1.20. So, the probability that you will win is 1/3 and the probability you will lose is 2/3. Your financial benefit will...
3. Roll a fair die 10 times. Call a number in 1, 2, 3, 4, 5, 6 a loner if it is rolled exactly once on the 10 rolls. (For example, if the rolls are 1 5 6 4 4 4 6 2 4 1, then 5 and 2 are the only loners) a. Compute the probability that at leas tone of numbers 1, 2, 3 is a loner.