Suppose you are rolling a fair four-sided die and a fair six-sided die and you are...

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.

a) Distinguish between the outcomes and events.

b) What is the probability that both die roll ones?

c) What is the probability that exactly one die rolls a one?

d) What is the probability that neither die rolls a one?

e) What is the expected number of ones?

f) If you did this 1000 times, approximately how many times would you expect that exactly one die would roll a one?

g) What is the expected number of ones?

Expert Solution

orchestra answered 2 years ago

Consider rolling both a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6...

Consider rolling both a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6 together. After rolling both dice, let X denote the number appearing on the foursided die and Y the number appearing on the six-sided die. Define W = X +Y . Assume X and Y are independent. (a) Find the moment generating function for W. (b) Use the moment generating function technique to find the expectation. (c) Use the moment generating function technique to find...

The experiment of rolling a fair six-sided die twice and looking at the values of the...

The experiment of rolling a fair six-sided die twice and looking at the values of the faces that are facing up, has the following sample space. For example, the result (1,2) implies that the face that is up from the first die shows the value 1 and the value of the face that is up from the second die is 2. sample space of tossing 2 die A pair of dice is thrown. Let X = the number of multiples...

1. The experiment of rolling a fair six-sided die twice and looking at the values of...

1. The experiment of rolling a fair six-sided die twice and looking at the values of the faces that are facing up, has the following sample space. For example, the result (1,2) implies that the face that is up from the first die shows the value 1 and the value of the face that is up from the second die is 2. (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)...

Suppose a red six-sided die, a blue six-sided die, and a yellow six-sided die are rolled....

Suppose a red six-sided die, a blue six-sided die, and a yellow six-sided die are rolled. Let - X1 be the random variable which is 1 if the numbers rolled on the blue die and the yellow die are the same and 0 otherwise; - X2 be the random variable which is 1 if the numbers rolled on the red die and the yellow die are the same and 0 otherwise; - X3 be the random variable which is 1...

Consider a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6, where X...

Consider a fair four-sided die numbered 1-4 and a fair six-sided die numbered 1-6, where X is the number appearing on the four-sided die and Y is the number appearing on the six-sided die. Define W=X+Y when they are rolled together. Assuming X and Y are independent, (a) find the moment generating function for W, (b) the expectation E(W), (c) and the variance Var(W). Use the moment generating function technique to find the expectation and variance.

Rolling a fair six-sided die five times could result in the following sample of n =...

Rolling a fair six-sided die five times could result in the following sample of n = 5 observations: What are the mean, variance, and standard deviation?

Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls...

Consider the experiment of rolling a six-sided fair die. Let X denote the number of rolls it takes to obtain the first 5, Y denote the number of rolls until the first 2, and Z denote the number of rolls until the first 4. Numerical answers are needed only for parts (a) and (b). Expressions are sufficient for parts (c), (d), and (e). a) E[X|Y = 1 or Z = 1] b) E[X|Y = 1 and Z = 2] c)...

You are interested in the probability of the outcome when rolling a six-sided die. Answer the...

You are interested in the probability of the outcome when rolling a six-sided die. Answer the following questions using p to denote the probability of this outcome. (a) You suspect that for this die the outcome does not occur with the same probability as with a fair die. Formulate the null and alternative hypotheses based on which you may make a statistical inference regarding your suspicion. (b) You suspect that for this die the outcome occurs with greater probability than...

You roll a fair six-sided die and don't look at it. What is the probability that...

You roll a fair six-sided die and don't look at it. What is the probability that it is a 5 given that your friend looks and tells you that it is greater than 2? Leave your answer as a fraction.

Let ? be the number that shows up when you roll a fair, six-sided die, and,...

Let ? be the number that shows up when you roll a fair, six-sided die, and, let ? = ?^2 − 5? + 6. a. Find both formats for the distribution of ?. (Hint: tep forms of probability distributions are CDF and pmf/pdf.) b. Find F(2.35)

Question