P(X) = 1/6 (for equally likely outcome)
E(X) = X * P(X)
= 1 * 1/6 + 2 * 1/6 + 3 * 1/6 + 4 * 1/6 + 5 * 1/6 + 6 * 1/6
= 3.5
Problem 6 (15 pts) Suppose we roll a die that is equally likely to have any...
1. Suppose we have a fair die that has twelve (12) sides. That is, if we roll it, each of the first 12 positive integers are equally likely to be the result of the roll. (a) If we roll the die, what is the probability the result is prime? As a reminder, one is not a prime number. (b) Suppose we roll this die 1000 times. What is the probability we get a prime number exactly 200 times? (c) Suppose...
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].
1) Suppose we have a fair 6 sided die and a coin. a) If we roll the die 4 times, the total number of possible outcomes is? b) If we roll the die 2 times then flip the coin 3 times, the total number of possible outcomes is? Show your calculations.
If we roll a red 6-sided die and a green 6-sided die (both are fair dice with the numbers 1-6 equally likely to be rolled), what is the probability that we get (i) A 5 on the green die AND a 3 on the red die? (ii) A 5 on the green die OR a 3 on the red die? (iii) A 5 on the green die GIVEN we rolled a 3 on the red die?
suppose you only have one fair 6-sided die. We will say that a success is if you roll a 5 or a 6. You roll the die over and over until you roll two successes in a row. What is the the expected number of times you must roll before you stop?
i. Consider a weighted 6-sided die that is twice as likely to produce any even outcome as any odd outcome. What is the expected value of 1 roll of this die? What is the expected value of the sum of 9 rolls of this die? ii. Let X denote the value of the sum of 10 rolls of an unweighted 6-sided die. What is Pr(X = 0 mod 6)? (Hint: it is sufficient to consider just the last roll) *side...
We flip a coin. If it is heads we roll a four sided die with sides numbered from 1 to 4. If it is tails, we roll a six sided die with sides numbered from 1 to 6. We let X be the number rolled. (a) What is the expectation of X? (b) What is the variance of X? (c) What is the standard deviation of X? We draw cards one by one and with replacement from a standard deck...
Problem 3 Roll a die until we get a 6. Let X be the total number of rolls and Y the number of l's we get. (a) Find Etx Y k (b) Find EY Problem 3 Roll a die until we get a 6. Let X be the total number of rolls and Y the number of l's we get. (a) Find Etx Y k (b) Find EY
1) Suppose you have a six-sided die. The die, unlike normal ones, has three sides with number 1, one side with number 2, and two sides with number 3. You roll this die once. Define the rauou ariable X to b ihe wing up afer the roll. a) List all possible outcomes of the random variable X and the corresponding probabilities b) Calculate the mean and the variance of the random variable. X
Question 3 3 pts Matching problem [Choose] You roll a fair six-sided die 500 times and observe a 3 on 90 of the 500 rolls. You estimate the probability of rolling a 3 to be 0.18 Choose) You roll a fair six-sided die 10 times and observe a 3 on all 10 rolls. You bet the probability of rolling a 3 on the next rollis close to O since you have already had 10 3's in a row You assign...