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Question 3 (14) Suppose that you repeatedly play a game where you roll a fair six-sided die. You win $10 if the number of dots faces up is 6. Otherwise, you win $1. Let $S_n$ be the total amount that you win after $n$ games and assume that you don't have any money in the beginning. (a) (2) Find $E[S_{10}]$ (b) (2) Find $VAR[S_{10}]$. (c) (2) Find $COV(S_{10}, S_{20})$. For the following questions (c) to (g), only give a clear mathematical expression and no need to provide a numerical answer. (d) (2) Find the probability that you win $28 in 10 games, $P[S_{10} = 28]$. (e) (2) Find $P[S_{30} = 120|S_{10} = 28]$. (f) (2) Find $P[S_{30} = 120 \cap S_{10} = 28]$. (g) (2) Assume that you now start out with $10 and each time you play the game costs you $4. The cost of playing the game is paid after completing 10 times. What is the probability that you have money larger than $5 after playing the game 10 times?

          Question 3 (14)
Suppose that you repeatedly play a game where you roll a fair six-sided die. You win $10 if the number of dots
faces up is 6. Otherwise, you win $1. Let $S_n$ be the total amount that you win after $n$ games and assume that
you don't have any money in the beginning.
(a) (2) Find $E[S_{10}]$
(b) (2) Find $VAR[S_{10}]$.
(c) (2) Find $COV(S_{10}, S_{20})$.
For the following questions (c) to (g), only give a clear mathematical expression and no need to provide a
numerical answer.
(d) (2) Find the probability that you win $28 in 10 games, $P[S_{10} = 28]$.
(e) (2) Find $P[S_{30} = 120|S_{10} = 28]$.
(f) (2) Find $P[S_{30} = 120 \cap S_{10} = 28]$.
(g) (2) Assume that you now start out with $10 and each time you play the game costs you $4. The cost of
playing the game is paid after completing 10 times. What is the probability that you have money larger than $5
after playing the game 10 times?

Question 3 (14)
Suppose that you repeatedly play a game where you roll a fair six-sided die. You win 10 if the number of dots
faces up is 6. Otherwise, you win1. Let Sn be the total amount that you win after n games and assume that
you don't have any money in the beginning.
(a) (2) Find E[S10]
(b) (2) Find VAR[S10].
(c) (2) Find COV(S10, S20).
For the following questions (c) to (g), only give a clear mathematical expression and no need to provide a
numerical answer.
(d) (2) Find the probability that you win 28 in 10 games,P[S10 = 28].
(e) (2) FindP[S30 = 120|S10 = 28].
(f) (2) FindP[S30 = 120 ∩S10 = 28].
(g) (2) Assume that you now start out with10 and each time you play the game costs you 4. The cost of
playing the game is paid after completing 10 times. What is the probability that you have money larger than5
after playing the game 10 times?

Added by Oscar L.

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