2. (13 pts) Suppose that you repeatedly play a game where you roll a fair six-sided die. If the number of dots faces up is 6, you win $2; otherwise, you win $0. Let $S_n$ be the total amount that you win after n games. Assume $S_0 = 0$. (a) Find $E[S_{15}]$, $Var[S_{15}]$, and $C_S(15, 20) = Cov(S_{15}, S_{20})$. (b) Find the probability that you win $4 in 15 games, $P[S_{15} = 4]$. (c) Find a mathematical expression for $P[S_{20} = 8 | S_{15} = 4]$. You do not need to evaluate it numerically. Just give an expression that could be evaluated numerically using Python or a calculator. (d) Find a mathematical expression for $P[S_{20} = 8 cap S_{15} = 4]$. (e) Now suppose that you start out with $5 ($S_0 = 5$), but that each time you play the game costs you $1. What is the probability that you still have money after playing the game 8 times ($S_8 > 0$)?
Added by Tammy R.
Close
Step 1
(a) Show moreā¦
Show all steps
Your feedback will help us improve your experience
Adi S and 94 other Intro Stats / AP Statistics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Suppose that you repeatedly play a game where you roll a fair six-sided die. If the number of dots faces up is 6, you win $2; otherwise, you win $0. Let Sn be the total amount that you win after n games. Assume S0 = 0. (a) Find E[S15], Var[S15], and Cov(S15, S20). (b) Find the probability that you win $4 in 15 games, P[S15 = 4]. (c) Find a mathematical expression for P[S20 = 8 | S15 = 4]. You do not need to evaluate it numerically. Just give an expression that could be evaluated numerically using Python or a calculator. (d) Find a mathematical expression for P[S20 = 8 ā© S15 = 4]. (e) Now suppose that you start out with $5 (S0 = 5), but that each time you play the game costs you $1. What is the probability that you still have money after playing the game 8 times (S8 > 0)?
Dominador T.
A four-sided die is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. a. Construct a probabilistic model for a single roll of this die and find the probability that the outcome is no larger than 3. b. You are offered the following game: The entrance fee is $1 and this die is rolled 576 times with the number of times of observing a number less than or equal to two counted. If this number is more than 312 or less than 264, you win $10. 1. What is the distribution that Y, the number of times of observing a number less than or equal to two, is following? 2. Write down the exact formula for your probability of winning and losing. 3. Make your decision on whether to play or not and justify your answer by approximating the expected winning per game.
Ameer S.
A four-sided die is loaded in a way that each even face is twice as likely as each odd face. All even faces are equally likely, as are all odd faces. a. Construct a probabilistic model for a single roll of this die and find the probability that the outcome is no larger than 3. b. You are offered the following game: The entrance fee is $1 and this die is rolled for 576 times with the number of times of observing a number less than or equal to two counted. If this number is more than 312 or less than 264, you win $10. What is the distribution that, Y = the number times of observing a number less than or equal to two, is following? winning per game.
Supreeta N.
Recommended Textbooks
Elementary Statistics a Step by Step Approach
The Practice of Statistics for AP
Introductory Statistics
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD