I need help with the questions in bold please. Thank you For the following probl
ID: 3889383 • Letter: I
Question
I need help with the questions in bold please. Thank you
For the following problem, clearly describe the sample space and the random variables you use. Be sure to justify where you get your expected values from. Consider playing a game where you roll n fair six-sided dice. For every 1 or 6 you roll you win $30, for rolling any other number you lose $3n.
(2) Now assume n = 6 (i.e., you roll six six-sided dice).
(a) (1 point) Describe the sample space for this experiment (you don’t need to list the elements but describe what is contained in it).
(b) (1 point) Describe a random variable which maps an outcome of this experiment to the winnings you receive. (Hint: Express your random variable as the sum of six random variables.)
(c) (1 point) Compute the expected value of this random variable using the linearity of expectation. Based on this would you play this game?
(3) (1 point) What is the largest value of n for which you would still want to play the game? Justify your answer.
Explanation / Answer
m(j) = P(T = j) = q j1 p .
Thus, E(T) = 1 · p + 2qp + 3q 2 p + · · · = p(1 + 2q + 3q 2 + · · ·)
. Now if |x| < 1, then 1 + x + x 2 + x 3 + · · · = 1 1 x .
Differentiating this formula, we get 1 + 2x + 3x 2 + · · · = 1 (1 x) 2 , so E(T) = p (1 q) 2 = p p 2 = 1 p .
In particular, we see that if we toss a fair coin a sequence of times, the expected time until the first heads is 1/(1/2) = 2. If we roll a die a sequence of times, the expected number of rolls until the first six is 1/(1/6) = 6.
P(x) * X.
X is the number of trials and P(x) is the probability of success. For example, if you toss a coin ten times, the probability of getting a heads in each trial is 1/2 so the expected value (the number of heads you can expect to get in 10 coin tosses) is:
P(x) * X = .5 * 10 = 5