(a). Roll a die until you get your 17th ace. Let T be the number of...

(a). Roll a die until you get your 17th ace. Let T be the number of rolls you need to get that 17th ace.

Find E(T) and var(T).

(b). Let Y = T-17 = "number of non-aces rolled to get your 17th ace". Y is called a "negative binomial" random variable, with parameters r=17 and p= 1/6.

Find E(Y) and var(Y).

(c). Find the approximate value of P{ T > 120 }

This variance can also be found in a tedious way (but not requiring any cleverness) using the "moment-generating function" of X given by m(t) = E[ e^(tX)]. Look it up if you have OCD.

Expert Solution

milcah answered 1 year ago

Suppose that you roll a die and your score is the number shown on the die....

Suppose that you roll a die and your score is the number shown on the die. On the other hand, suppose that your friend rolls five dice and his score is the number of 6’s shown out of five rollings. Compute the probability (a) that the two scores are equal. (b) that your friend’s score is strictly smaller than yours.

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)

(a) If you roll a single die and count the number of dots on top, what...

(a) If you roll a single die and count the number of dots on top, what is the sample space of all possible outcomes? Are the outcomes equally likely? (b) Assign probabilities to the outcomes of the sample space of part (a). Do the probabilities add up to 1? Should they add up to 1? Explain. (c) What is the probability of getting a number less than 5 on a single throw? (d) What is the probability of getting 5...

You roll a die, winning nothing if the number of spots is even, $3 for a...

You roll a die, winning nothing if the number of spots is even, $3 for a 1 or a 3, and $18 for a 5. a) Find the expected value and standard deviation of your prospective winnings. b) You play three times. Find the mean and standard deviation of your total winnings. c) You play 30 times. What is the probability that you win at least $160?

You roll a die, winning nothing if the number of spots is eveneven, $77 for a...

You roll a die, winning nothing if the number of spots is eveneven, $77 for a 11 or a 33, and $1616 for a 55. a) Find the expected value and standard deviation of your prospective winnings. b) You play three timesthree times. Find the mean and standard deviation of your total winnings. c) You play 3030 times. What is the probability that you win at least $190190?

What is the probability that, if you roll a balanced die twice, that you will get a "1" on both dice?

Consider the following random variables based on a roll of a fair die, (1) You get...

Consider the following random variables based on a roll of a fair die, (1) You get $0.5 multiplied by the outcome of the die (2) You get $0 if the outcome is below or equal 3 and $1 otherwise (3) You get $1 if the outcome is below or equal 3 and $0 otherwise (4) You pay $1.5 if the outcome is below or equal 3 and get $1 multiplied the outcome otherwise Die Outcome Random Variable (1) Random Variable...

You roll one blue 6-sided die, and one red 4-sided die. Let A be the event...

You roll one blue 6-sided die, and one red 4-sided die. Let A be the event that the outcome of the red die is twice the outcome on the blue die Let B be the event that the outcome of the blue die is greater than the outcome on the red die Let C be the event that the sum of the two dice is a prime number. a) Find P(B) and P(C) by clearly listing all outcomes in B...

A Betting Game You roll a die. If an odd number comes up, you lose. If...

A Betting Game You roll a die. If an odd number comes up, you lose. If you get a 6, you win $60. If it is an even number other than 6, you get to roll again. If you get a 6 the second time, you win $36. If not, you lose. (a) Construct a probability model for the amount you win at this game. Explain briefly how you obtain the probabilities associated with the different amounts of winning. (b)...

You roll a balanced die two times. Is the second roll independent of the first roll?

Question