Question

A gambler has the following strategy when playing the roulette: ? she bets 1 dollar on red and if she wins she stops, ? if she loses, she bets 2 dollars on red and then stops. The roulette has 38 numbers from which 0 and 00 are green and the rest are red or black. When a gambler bets on red, the gambler wins if the ball stops at any of the red numbers. In addition, the payout when betting on colors is 1:1 (i.e. if one wins, one gets back double the money one bet). a) Find the probability she loses 3 dollars. b) Find the probability she won on the first bet given that she made 1 dollar. c) If X is the amount of money she won, find E[X]. d) Suppose the gambler changes her strategy and now bets 1 dollar every time on red and never stops. So, on every bet she either wins or loses a dollar. Find the expected number of bets she will make till she wins at least 3 out of 4 consecutive bets. 

          A gambler has the following strategy when playing the roulette:

? she bets 1 dollar on red and if she wins she stops,

? if she loses, she bets 2 dollars on red and then stops.

The roulette has 38 numbers from which 0 and 00 are green and the rest are red or black. When a gambler bets on red, the gambler wins if the ball stops at any of the red numbers. In addition, the payout when betting on colors is 1:1 (i.e. if one wins, one gets back double the money one bet).

a) Find the probability she loses 3 dollars.

b) Find the probability she won on the first bet given that she made 1 dollar.

c) If X is the amount of money she won, find E[X].

d) Suppose the gambler changes her strategy and now bets 1 dollar every time on red and never stops. So, on every bet she either wins or loses a dollar. Find the expected number of bets she will make till she wins at least 3 out of 4 consecutive bets.
Show more…
A gambler has the following strategy when playing the roulette: ? she bets 1 dollar on red and if she wins she stops, ? if she loses, she bets 2 dollars on red and then stops. The roulette has 38 numbers from which 0 and 00 are green and the rest are red or black. When a gambler bets on red, the gambler wins if the ball stops at any of the red numbers. In addition, the payout when betting on colors is 1:1 (i.e. if one wins, one gets back double the money one bet). a) Find the probability she loses 3 dollars. b) Find the probability she won on the first bet given that she made 1 dollar. c) If X is the amount of money she won, find E[X]. d) Suppose the gambler changes her strategy and now bets 1 dollar every time on red and never stops. So, on every bet she either wins or loses a dollar. Find the expected number of bets she will make till she wins at least 3 out of 4 consecutive bets.

Added by Michael K.

Close

Elementary Statistics a Step by Step Approach
Elementary Statistics a Step by Step Approach
Allan G. Bluman 9th Edition
AceChat toggle button
Close icon
Ace pointing down

Please give Ace some feedback

Your feedback will help us improve your experience

Thumb up icon Thumb down icon
Thanks for your feedback!
Profile picture
A gambler has the following strategy when playing the roulette: * she bets 1 dollar on red and if she wins she stops, * if she loses, she bets 2 dollars on red and then stops. The roulette has 38 numbers from which 0 and 00 are green and the rest are red or black. When a gambler bets on red, the gambler wins if the ball stops at any of the red numbers. In addition, the payout when betting on colors is 1:1 (i.e. if one wins, one gets back double the money one bet). a) Find the probability she loses 3 dollars. b) Find the probability she won on the first bet given that she made 1 dollar. c) If X is the amount of money she won, find E[X]. d) Suppose the gambler changes her strategy and now bets 1 dollar every time on red and never stops. So, on every bet she either wins or loses a dollar. Find the expected number of bets she will make till she wins at least 3 out of 4 consecutive bets.
Close icon
Play audio
Feedback
Powered by NumerAI
Danielle Fairburn David Collins
Jennifer Stoner verified

Rashmi Sinha and 54 other subject Intro Stats / AP Statistics educators are ready to help you.

Ask a new question
*

Labs

-

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs
*

Key Concepts

-
Key Concept
Premium Feature
Explore the core concept behind this problem.
Play button
Key Concept
Premium Feature
Explore the core concept behind this problem.
Your browser does not support the video tag.
*

Recommended Videos

-
a-roulette-wheel-has-38-slots-numbered-0-00-and-1-to-36-the-slots-0-and-00-are-colored-green-18-of-the-others-are-red-and-18-are-blackthe-dealer-spins-the-wheel-and-at-the-same-time-rolls-a-09001

A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so that the ball is equally likely to land in any slot when the wheel slows. Gamblers can bet on various combinations of numbers and colors. (a) If you bet on "red," you win if the ball lands in a red slot. What is the probability of winning with a bet on red in a single play of roulette? (b) You decide to play roulette four times, each time betting on red. What is the distribution of X, the number of times you win? (c) If you bet the same amount on each play and win on exactly four of the eight plays, then you will "break even." What is the probability that you will break even? (d) If you win on fewer than four of the eight plays, then you will lose money. What is the probability that you will lose money?

Shaiju T.
a-roulette-wheel-has-38-slots-of-which-18-are-black-18-are-red-and-2-are-green-when-the-wheel-is-spun-the-ball-is-equally-likely-to-come-to-rest-in-any-of-the-slots-one-of-the-simplest-wager-22595

A roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers chooses red or black. A bet of $1 on red returns $2 if the ball lands in a red slot. Otherwise, the player loses his dollar. When gamblers bet on red or black, the two green slots belong to the house. Because the probability of winning $2 is 18 38 , the mean payoff from a $1 bet is twice 18 38 , or 94.7 cents. What does the law of large numbers say will happen if a gambler bets on red over and over again? A. In the long term, the gambler will earn an average of 94.7 cents per $1 bet. B. The gambler can improve their odds at winning by changing their bets to black. C. The gambler must win in the long haul if they keep playing. D. On average, the gambler is going to lose 94.7 cents per bet.

Maitreya T.
a-roulette-wheel-has-38-slots-of-which-18-are-black-18-are-red-and-2-are-green-when-the-wheel-is-spun-the-ball-is-equally-likely-to-come-to-rest-in-any-of-the-slots-one-of-the-simplest-wager-22595

A roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. One of the simplest wagers chooses red or black. A bet of $1 on red returns $2 if the ball lands in a red slot. Otherwise, the player loses his dollar. When gamblers bet on red or black, the two green slots belong to the house. Because the probability of winning $2 is 18 38 , the mean payoff from a $1 bet is twice 18 38 , or 94.7 cents. What does the law of large numbers say will happen if a gambler bets on red over and over again? A. In the long term, the gambler will earn an average of 94.7 cents per $1 bet. B. The gambler can improve their odds at winning by changing their bets to black. C. The gambler must win in the long haul if they keep playing. D. On average, the gambler is going to lose 94.7 cents per bet.

Maitreya T.
*

Recommended Textbooks

-
Elementary Statistics a Step by Step Approach

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition
achievement 1,899 solutions
The Practice of Statistics for AP

The Practice of Statistics for AP

Daren S. Starnes, Daniel S. Yates, David S. Moore 4th Edition
achievement 1,228 solutions
Introductory Statistics

Introductory Statistics

Barbara Illowsky, Susan Dean 1st Edition
achievement 1,393 solutions
*

Transcript

-
00:01 So there are total 38 numbers of which two are green.
00:05 18 are red and 18 are black.
00:08 Let the probability that ball stops at red be.
00:14 Ball stops at red be r.
00:21 So r is 18 upon total is 38.
00:26 That is 9 upon 19.
00:29 So first, the probability that she loses three dollars so to lose three dollars to lose three dollars she should lose should lose the first game the first game and then she should also lose in the second and then stops the game so the probability that she loses the game is 1 minus r hence the probability is 1 minus r that is 9 upon 19 multiplied by 9 upon 19 which is 0 .2244.
01:42 Second the probability that she won on the first bet given she made one dollar let why let why be the event that she made one dollar so she won the first game and stops so wins one dollars and she loses the first game she wins first game she wins first game and she makes one dollar and she loses first game and wins second so she makes, loses $1 and wins two...
Need help? Use Ace
Ace is your personal tutor. It breaks down any question with clear steps so you can learn.
Start Using Ace
Ace is your personal tutor for learning
Step-by-step explanations
Instant summaries
Summarize YouTube videos
Understand textbook images or PDFs
Study tools like quizzes and flashcards
Listen to your notes as a podcast
Continue solving this problem
Create a free account to:
  • View full step-by-step solution
  • Ask follow-up questions with Ace AI
  • Save progress and study later
Continue Free
Numerade

Get step-by-step video solution
from top educators

Continue with Clever
or



By creating an account, you agree to the Terms of Service and Privacy Policy
Already have an account? Log In

A free answer
just for you

Watch the video solution with this free unlock.

Numerade

Log in to watch this video
...and 100,000,000 more!


EMAIL

PASSWORD

OR
Continue with Clever